The document describes polar coordinates, which represent the location of a point P in a plane using two numbers: r, the distance from P to the origin O, and θ, the angle between the positive x-axis and the line from O to P. θ is positive for counter-clockwise angles and negative for clockwise angles. The polar coordinate (r, θ) uniquely identifies P's location. The document also provides the conversion formulas between polar coordinates (r, θ) and rectangular coordinates (x, y).
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
The Mean Value Theorem is the Most Important Theorem in Calculus. It allows us to relate information about the derivative of a function to information about the function itself.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
The Mean Value Theorem is the Most Important Theorem in Calculus. It allows us to relate information about the derivative of a function to information about the function itself.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at Uni...Don Sheehy
The Vietoris-Rips filtration is a versatile tool in topological data analysis.
Unfortunately, it is often too large to construct in full.
We show how to construct an $O(n)$-size filtered simplicial complex on an $n$-point metric space such that the persistence diagram is a good approximation to that of the Vietoris-Rips filtration.
The filtration can be constructed in $O(n\log n)$ time.
The constants depend only on the doubling dimension of the metric space and the desired tightness of the approximation.
For the first time, this makes it computationally tractable to approximate the persistence diagram of the Vietoris-Rips filtration across all scales for large data sets.
Our approach uses a hierarchical net-tree to sparsify the filtration.
We can either sparsify the data by throwing out points at larger scales to give a zigzag filtration,
or sparsify the underlying graph by throwing out edges at larger scales to give a standard filtration.
Both methods yield the same guarantees.
Meet Crazyjamjam - A TikTok Sensation | Blog EternalBlog Eternal
Crazyjamjam, the TikTok star everyone's talking about! Uncover her secrets to success, viral trends, and more in this exclusive feature on Blog Eternal.
Source: https://blogeternal.com/celebrity/crazyjamjam-leaks/
Scandal! Teasers June 2024 on etv Forum.co.zaIsaac More
Monday, 3 June 2024
Episode 47
A friend is compelled to expose a manipulative scheme to prevent another from making a grave mistake. In a frantic bid to save Jojo, Phakamile agrees to a meeting that unbeknownst to her, will seal her fate.
Tuesday, 4 June 2024
Episode 48
A mother, with her son's best interests at heart, finds him unready to heed her advice. Motshabi finds herself in an unmanageable situation, sinking fast like in quicksand.
Wednesday, 5 June 2024
Episode 49
A woman fabricates a diabolical lie to cover up an indiscretion. Overwhelmed by guilt, she makes a spontaneous confession that could be devastating to another heart.
Thursday, 6 June 2024
Episode 50
Linda unwittingly discloses damning information. Nhlamulo and Vuvu try to guide their friend towards the right decision.
Friday, 7 June 2024
Episode 51
Jojo's life continues to spiral out of control. Dintle weaves a web of lies to conceal that she is not as successful as everyone believes.
Monday, 10 June 2024
Episode 52
A heated confrontation between lovers leads to a devastating admission of guilt. Dintle's desperation takes a new turn, leaving her with dwindling options.
Tuesday, 11 June 2024
Episode 53
Unable to resort to violence, Taps issues a verbal threat, leaving Mdala unsettled. A sister must explain her life choices to regain her brother's trust.
Wednesday, 12 June 2024
Episode 54
Winnie makes a very troubling discovery. Taps follows through on his threat, leaving a woman reeling. Layla, oblivious to the truth, offers an incentive.
Thursday, 13 June 2024
Episode 55
A nosy relative arrives just in time to thwart a man's fatal decision. Dintle manipulates Khanyi to tug at Mo's heartstrings and get what she wants.
Friday, 14 June 2024
Episode 56
Tlhogi is shocked by Mdala's reaction following the revelation of their indiscretion. Jojo is in disbelief when the punishment for his crime is revealed.
Monday, 17 June 2024
Episode 57
A woman reprimands another to stay in her lane, leading to a damning revelation. A man decides to leave his broken life behind.
Tuesday, 18 June 2024
Episode 58
Nhlamulo learns that due to his actions, his worst fears have come true. Caiphus' extravagant promises to suppliers get him into trouble with Ndu.
Wednesday, 19 June 2024
Episode 59
A woman manages to kill two birds with one stone. Business doom looms over Chillax. A sobering incident makes a woman realize how far she's fallen.
Thursday, 20 June 2024
Episode 60
Taps' offer to help Nhlamulo comes with hidden motives. Caiphus' new ideas for Chillax have MaHilda excited. A blast from the past recognizes Dintle, not for her newfound fame.
Friday, 21 June 2024
Episode 61
Taps is hungry for revenge and finds a rope to hang Mdala with. Chillax's new job opportunity elicits mixed reactions from the public. Roommates' initial meeting starts off on the wrong foot.
Monday, 24 June 2024
Episode 62
Taps seizes new information and recruits someone on the inside. Mary's new job
Maximizing Your Streaming Experience with XCIPTV- Tips for 2024.pdfXtreame HDTV
In today’s digital age, streaming services have become an integral part of our entertainment lives. Among the myriad of options available, XCIPTV stands out as a premier choice for those seeking seamless, high-quality streaming. This comprehensive guide will delve into the features, benefits, and user experience of XCIPTV, illustrating why it is a top contender in the IPTV industry.
Panchayat Season 3 - Official Trailer.pdfSuleman Rana
The dearest series "Panchayat" is set to make a victorious return with its third season, and the fervor is discernible. The authority trailer, delivered on May 28, guarantees one more enamoring venture through the country heartland of India.
Jitendra Kumar keeps on sparkling as Abhishek Tripathi, the city-reared engineer who ends up functioning as the secretary of the Panchayat office in the curious town of Phulera. His nuanced depiction of a young fellow exploring the difficulties of country life while endeavoring to adjust to his new environmental factors has earned far and wide recognition.
Neena Gupta and Raghubir Yadav return as Manju Devi and Brij Bhushan Dubey, separately. Their dynamic science and immaculate acting rejuvenate the hardships of town administration. Gupta's depiction of the town Pradhan with an ever-evolving outlook, matched with Yadav's carefully prepared exhibition, adds profundity and credibility to the story.
New Difficulties and Experiences
The trailer indicates new difficulties anticipating the characters, as Abhishek keeps on wrestling with his part in the town and his yearnings for a superior future. The series has reliably offset humor with social editorial, and Season 3 looks ready to dig much more profound into the intricacies of rustic organization and self-awareness.
Watchers can hope to see a greater amount of the enchanting and particular residents who have become fan top picks. Their connections and the one of a kind cut of-life situations give a reviving and interesting portrayal of provincial India, featuring the two its appeal and its difficulties.
A Mix of Humor and Heart
One of the signs of "Panchayat" is its capacity to mix humor with sincere narrating. The trailer features minutes that guarantee to convey giggles, as well as scenes that pull at the heartstrings. This equilibrium has been a critical calculate the show's prosperity, resounding with crowds across different socioeconomics.
Creation Greatness
The creation quality remaining parts first rate, with the beautiful setting of Phulera town filling in as a scenery that upgrades the narrating. The meticulousness in portraying provincial life, joined with sharp composition and solid exhibitions, guarantees that "Panchayat" keeps on hanging out in the packed web series scene.
Expectation and Delivery
As the delivery date draws near, expectation for "Panchayat" Season 3 is at a record-breaking high. The authority trailer has previously created critical buzz, with fans enthusiastically anticipating the continuation of Abhishek Tripathi's excursion and the new undertakings that lie ahead in Phulera.
All in all, the authority trailer for "Panchayat" Season 3 recommends that watchers are in for another drawing in and engaging ride. Yet again with its charming characters, convincing story, and ideal mix of humor and show, the new season is set to enamor crowds. Write in your schedules and prepare to get back to the endearing universe of "Panchayat."
As a film director, I have always been awestruck by the magic of animation. Animation, a medium once considered solely for the amusement of children, has undergone a significant transformation over the years. Its evolution from a rudimentary form of entertainment to a sophisticated form of storytelling has stirred my creativity and expanded my vision, offering limitless possibilities in the realm of cinematic storytelling.
Skeem Saam in June 2024 available on ForumIsaac More
Monday, June 3, 2024 - Episode 241: Sergeant Rathebe nabs a top scammer in Turfloop. Meikie is furious at her uncle's reaction to the truth about Ntswaki.
Tuesday, June 4, 2024 - Episode 242: Babeile uncovers the truth behind Rathebe’s latest actions. Leeto's announcement shocks his employees, and Ntswaki’s ordeal haunts her family.
Wednesday, June 5, 2024 - Episode 243: Rathebe blocks Babeile from investigating further. Melita warns Eunice to stay clear of Mr. Kgomo.
Thursday, June 6, 2024 - Episode 244: Tbose surrenders to the police while an intruder meddles in his affairs. Rathebe's secret mission faces a setback.
Friday, June 7, 2024 - Episode 245: Rathebe’s antics reach Kganyago. Tbose dodges a bullet, but a nightmare looms. Mr. Kgomo accuses Melita of witchcraft.
Monday, June 10, 2024 - Episode 246: Ntswaki struggles on her first day back at school. Babeile is stunned by Rathebe’s romance with Bullet Mabuza.
Tuesday, June 11, 2024 - Episode 247: An unexpected turn halts Rathebe’s investigation. The press discovers Mr. Kgomo’s affair with a young employee.
Wednesday, June 12, 2024 - Episode 248: Rathebe chases a criminal, resorting to gunfire. Turf High is rife with tension and transfer threats.
Thursday, June 13, 2024 - Episode 249: Rathebe traps Kganyago. John warns Toby to stop harassing Ntswaki.
Friday, June 14, 2024 - Episode 250: Babeile is cleared to investigate Rathebe. Melita gains Mr. Kgomo’s trust, and Jacobeth devises a financial solution.
Monday, June 17, 2024 - Episode 251: Rathebe feels the pressure as Babeile closes in. Mr. Kgomo and Eunice clash. Jacobeth risks her safety in pursuit of Kganyago.
Tuesday, June 18, 2024 - Episode 252: Bullet Mabuza retaliates against Jacobeth. Pitsi inadvertently reveals his parents’ plans. Nkosi is shocked by Khwezi’s decision on LJ’s future.
Wednesday, June 19, 2024 - Episode 253: Jacobeth is ensnared in deceit. Evelyn is stressed over Toby’s case, and Letetswe reveals shocking academic results.
Thursday, June 20, 2024 - Episode 254: Elizabeth learns Jacobeth is in Mpumalanga. Kganyago's past is exposed, and Lehasa discovers his son is in KZN.
Friday, June 21, 2024 - Episode 255: Elizabeth confirms Jacobeth’s dubious activities in Mpumalanga. Rathebe lies about her relationship with Bullet, and Jacobeth faces theft accusations.
Monday, June 24, 2024 - Episode 256: Rathebe spies on Kganyago. Lehasa plans to retrieve his son from KZN, fearing what awaits.
Tuesday, June 25, 2024 - Episode 257: MaNtuli fears for Kwaito’s safety in Mpumalanga. Mr. Kgomo and Melita reconcile.
Wednesday, June 26, 2024 - Episode 258: Kganyago makes a bold escape. Elizabeth receives a shocking message from Kwaito. Mrs. Khoza defends her husband against scam accusations.
Thursday, June 27, 2024 - Episode 259: Babeile's skillful arrest changes the game. Tbose and Kwaito face a hostage crisis.
Friday, June 28, 2024 - Episode 260: Two women face the reality of being scammed. Turf is rocked by breaking
Hollywood Actress - The 250 hottest galleryZsolt Nemeth
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From the Editor's Desk: 115th Father's day Celebration - When we see Father's day in Hindu context, Nanda Baba is the most vivid figure which comes to the mind. Nanda Baba who was the foster father of Lord Krishna is known to provide love, care and affection to Lord Krishna and Balarama along with his wife Yashoda; Letter’s to the Editor: Mother's Day - Mother is a precious life for their children. Mother is life breath for her children. Mother's lap is the world happiness whose debt can never be paid.
Tom Selleck Net Worth: A Comprehensive Analysisgreendigital
Over several decades, Tom Selleck, a name synonymous with charisma. From his iconic role as Thomas Magnum in the television series "Magnum, P.I." to his enduring presence in "Blue Bloods," Selleck has captivated audiences with his versatility and charm. As a result, "Tom Selleck net worth" has become a topic of great interest among fans. and financial enthusiasts alike. This article delves deep into Tom Selleck's wealth, exploring his career, assets, endorsements. and business ventures that contribute to his impressive economic standing.
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Early Life and Career Beginnings
The Foundation of Tom Selleck's Wealth
Born on January 29, 1945, in Detroit, Michigan, Tom Selleck grew up in Sherman Oaks, California. His journey towards building a large net worth began with humble origins. , Selleck pursued a business administration degree at the University of Southern California (USC) on a basketball scholarship. But, his interest shifted towards acting. leading him to study at the Hills Playhouse under Milton Katselas.
Minor roles in television and films marked Selleck's early career. He appeared in commercials and took on small parts in T.V. series such as "The Dating Game" and "Lancer." These initial steps, although modest. laid the groundwork for his future success and the growth of Tom Selleck net worth. Breakthrough with "Magnum, P.I."
The Role that Defined Tom Selleck's Career
Tom Selleck's breakthrough came with the role of Thomas Magnum in the CBS television series "Magnum, P.I." (1980-1988). This role made him a household name and boosted his net worth. The series' popularity resulted in Selleck earning large salaries. leading to financial stability and increased recognition in Hollywood.
"Magnum P.I." garnered high ratings and critical acclaim during its run. Selleck's portrayal of the charming and resourceful private investigator resonated with audiences. making him one of the most beloved television actors of the 1980s. The success of "Magnum P.I." played a pivotal role in shaping Tom Selleck net worth, establishing him as a major star.
Film Career and Diversification
Expanding Tom Selleck's Financial Portfolio
While "Magnum, P.I." was a cornerstone of Selleck's career, he did not limit himself to television. He ventured into films, further enhancing Tom Selleck net worth. His filmography includes notable movies such as "Three Men and a Baby" (1987). which became the highest-grossing film of the year, and its sequel, "Three Men and a Little Lady" (1990). These box office successes contributed to his wealth.
Selleck's versatility allowed him to transition between genres. from comedies like "Mr. Baseball" (1992) to westerns such as "Quigley Down Under" (1990). This diversification showcased his acting range. and provided many income streams, reinforcing Tom Selleck net worth.
Television Resurgence with "Blue Bloods"
Sustaining Wealth through Consistent Success
In 2010, Tom Selleck began starring as Frank Reagan i
240529_Teleprotection Global Market Report 2024.pdfMadhura TBRC
The teleprotection market size has grown
exponentially in recent years. It will grow from
$21.92 billion in 2023 to $28.11 billion in 2024 at a
compound annual growth rate (CAGR) of 28.2%. The
teleprotection market size is expected to see
exponential growth in the next few years. It will grow
to $70.77 billion in 2028 at a compound annual
growth rate (CAGR) of 26.0%.
Meet Dinah Mattingly – Larry Bird’s Partner in Life and Loveget joys
Get an intimate look at Dinah Mattingly’s life alongside NBA icon Larry Bird. From their humble beginnings to their life today, discover the love and partnership that have defined their relationship.
Experience the thrill of Progressive Puzzle Adventures, like Scavenger Hunt Games and Escape Room Activities combined Solve Treasure Hunt Puzzles online.
In the vast landscape of cinema, stories have been told, retold, and reimagined in countless ways. At the heart of this narrative evolution lies the concept of a "remake". A successful remake allows us to revisit cherished tales through a fresh lens, often reflecting a different era's perspective or harnessing the power of advanced technology. Yet, the question remains, what makes a remake successful? Today, we will delve deeper into this subject, identifying the key ingredients that contribute to the success of a remake.
Create a Seamless Viewing Experience with Your Own Custom OTT Player.pdfGenny Knight
As the popularity of online streaming continues to rise, the significance of providing outstanding viewing experiences cannot be emphasized enough. Tailored OTT players present a robust solution for service providers aiming to enhance their offerings and engage audiences in a competitive market. Through embracing customization, companies can craft immersive, individualized experiences that effectively hold viewers' attention, entertain them, and encourage repeat usage.
From Slave to Scourge: The Existential Choice of Django Unchained. The Philos...Rodney Thomas Jr
#SSAPhilosophy #DjangoUnchained #DjangoFreeman #ExistentialPhilosophy #Freedom #Identity #Justice #Courage #Rebellion #Transformation
Welcome to SSA Philosophy, your ultimate destination for diving deep into the profound philosophies of iconic characters from video games, movies, and TV shows. In this episode, we explore the powerful journey and existential philosophy of Django Freeman from Quentin Tarantino’s masterful film, "Django Unchained," in our video titled, "From Slave to Scourge: The Existential Choice of Django Unchained. The Philosophy of Django Freeman!"
From Slave to Scourge: The Existential Choice of Django Unchained – The Philosophy of Django Freeman!
Join me as we delve into the existential philosophy of Django Freeman, uncovering the profound lessons and timeless wisdom his character offers. Through his story, we find inspiration in the power of choice, the quest for justice, and the courage to defy oppression. Django Freeman’s philosophy is a testament to the human spirit’s unyielding drive for freedom and justice.
Don’t forget to like, comment, and subscribe to SSA Philosophy for more in-depth explorations of the philosophies behind your favorite characters. Hit the notification bell to stay updated on our latest videos. Let’s discover the principles that shape these icons and the profound lessons they offer.
Django Freeman’s story is one of the most compelling narratives of transformation and empowerment in cinema. A former slave turned relentless bounty hunter, Django’s journey is not just a physical liberation but an existential quest for identity, justice, and retribution. This video delves into the core philosophical elements that define Django’s character and the profound choices he makes throughout his journey.
Link to video: https://youtu.be/GszqrXk38qk
Young Tom Selleck: A Journey Through His Early Years and Rise to Stardomgreendigital
Introduction
When one thinks of Hollywood legends, Tom Selleck is a name that comes to mind. Known for his charming smile, rugged good looks. and the iconic mustache that has become synonymous with his persona. Tom Selleck has had a prolific career spanning decades. But, the journey of young Tom Selleck, from his early years to becoming a household name. is a story filled with determination, talent, and a touch of luck. This article delves into young Tom Selleck's life, background, early struggles. and pivotal moments that led to his rise in Hollywood.
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Early Life and Background
Family Roots and Childhood
Thomas William Selleck was born in Detroit, Michigan, on January 29, 1945. He was the second of four children in a close-knit family. His father, Robert Dean Selleck, was a real estate investor and executive. while his mother, Martha Selleck, was a homemaker. The Selleck family relocated to Sherman Oaks, California. when Tom was a child, setting the stage for his future in the entertainment industry.
Education and Early Interests
Growing up, young Tom Selleck was an active and athletic child. He attended Grant High School in Van Nuys, California. where he excelled in sports, particularly basketball. His tall and athletic build made him a standout player, and he earned a basketball scholarship to the University of Southern California (U.S.C.). While at U.S.C., Selleck studied business administration. but his interests shifted toward acting.
Discovery of Acting Passion
Tom Selleck's journey into acting was serendipitous. During his time at U.S.C., a drama coach encouraged him to try acting. This nudge led him to join the Hills Playhouse, where he began honing his craft. Transitioning from an aspiring athlete to an actor took time. but young Tom Selleck became drawn to the performance world.
Early Career Struggles
Breaking Into the Industry
The path to stardom was a challenging one for young Tom Selleck. Like many aspiring actors, he faced many rejections and struggled to find steady work. A series of minor roles and guest appearances on television shows marked his early career. In 1965, he debuted on the syndicated show "The Dating Game." which gave him some exposure but did not lead to immediate success.
The Commercial Breakthrough
During the late 1960s and early 1970s, Selleck began appearing in television commercials. His rugged good looks and charismatic presence made him a popular brand choice. He starred in advertisements for Pepsi-Cola, Revlon, and Close-Up toothpaste. These commercials provided financial stability and helped him gain visibility in the industry.
Struggling Actor in Hollywood
Despite his success in commercials. breaking into large acting roles remained a challenge for young Tom Selleck. He auditioned and took on small parts in T.V. shows and movies. Some of his early television appearances included roles in popular series like Lancer, The F.B.I., and Bracken's World. But, it would take a
3. Polar Coordinates
The location of a point P in the plane may be given
by the following two numbers:
r = the distance between P and the origin O(0, 0)
y
P (r, θ)
r
x
O
4. Polar Coordinates
The location of a point P in the plane may be given
by the following two numbers:
r = the distance between P and the origin O(0, 0)
θ = a signed angle between the positive x–axis and
the direction to P,
y
P (r, θ)
r
θ
x
O
5. Polar Coordinates
The location of a point P in the plane may be given
by the following two numbers:
r = the distance between P and the origin O(0, 0)
θ = a signed angle between the positive x–axis and
the direction to P, specifically,
θ is + for counter clockwise measurements and
θ is – for clockwise measurements .
y
P (r, θ)
r
θ
x
O
6. Polar Coordinates
The location of a point P in the plane may be given
by the following two numbers:
r = the distance between P and the origin O(0, 0)
θ = a signed angle between the positive x–axis and
the direction to P, specifically,
θ is + for counter clockwise measurements and
θ is – for clockwise measurements
The ordered pair (r, θ) is a polar coordinate of P.
y
P (r, θ)
r
θ
x
O
7. Polar Coordinates
The location of a point P in the plane may be given
by the following two numbers:
r = the distance between P and the origin O(0, 0)
θ = a signed angle between the positive x–axis and
the direction to P, specifically,
θ is + for counter clockwise measurements and
θ is – for clockwise measurements
The ordered pair (r, θ) is a polar coordinate of P.
The ordered pairs (r, θ ±2nπ ) with y
n = 0,1, 2, 3… give the same P (r, θ)
geometric information hence lead to
the same location P(r, θ).
r
θ
x
O
8. Polar Coordinates
The location of a point P in the plane may be given
by the following two numbers:
r = the distance between P and the origin O(0, 0)
θ = a signed angle between the positive x–axis and
the direction to P, specifically,
θ is + for counter clockwise measurements and
θ is – for clockwise measurements
The ordered pair (r, θ) is a polar coordinate of P.
The ordered pairs (r, θ ±2nπ ) with y
n = 0,1, 2, 3… give the same P (r, θ)
geometric information hence lead to
the same location P(r, θ).
r
We also use signed distance,
i.e. with negative values of r which
θ
means we are to step backward for x
O
a distance of lrl.
9. Polar Coordinates
If needed, we write (a, b)P for a polar coordinate
ordered pair, and (a, b)R for the rectangular
coordinate ordered pair.
10. Polar Coordinates
If needed, we write (a, b)P for a polar coordinate
ordered pair, and (a, b)R for rectangular coordinate
ordered pair.
Conversion Rules
11. Polar Coordinates
If needed, we write (a, b)P for a polar coordinate
ordered pair, and (a, b)R for rectangular coordinate
ordered pair.
Conversion Rules
Let (x, y)R and (r, θ)P be the rectangular and polar
coordinates of the same point P, then
x= y
(r, θ) = (x, y)
R P
y= P
r=
r
θ x
O
The rectangular and polar
coordinates relations
12. Polar Coordinates
If needed, we write (a, b)P for a polar coordinate
ordered pair, and (a, b)R for rectangular coordinate
ordered pair.
Conversion Rules
Let (x, y)R and (r, θ)P be the rectangular and polar
coordinates of the same point P, then
x = r*cos(θ) y
(r, θ) = (x, y)
R P
y = r*sin(θ) P
r=
r
y = r*sin(θ)
θ x
O x = r*cos(θ)
The rectangular and polar
coordinates relations
13. Polar Coordinates
If needed, we write (a, b)P for a polar coordinate
ordered pair, and (a, b)R for rectangular coordinate
ordered pair.
Conversion Rules
Let (x, y)R and (r, θ)P be the rectangular and polar
coordinates of the same point P, then
x = r*cos(θ) y
(r, θ) = (x, y)
R P
y = r*sin(θ) P
r = √ x2 + y2
r
y = r*sin(θ)
θ x
O x = r*cos(θ)
The rectangular and polar
coordinates relations
14. Polar Coordinates
If needed, we write (a, b)P for a polar coordinate
ordered pair, and (a, b)R for rectangular coordinate
ordered pair.
Conversion Rules
Let (x, y)R and (r, θ)P be the rectangular and polar
coordinates of the same point P, then
x = r*cos(θ) y
(r, θ) = (x, y)
R P
y = r*sin(θ) P
r = √ x2 + y2
For θ we have r
y = r*sin(θ)
tan(θ) =
cos(θ) = θ x
O x = r*cos(θ)
The rectangular and polar
coordinates relations
15. Polar Coordinates
If needed, we write (a, b)P for a polar coordinate
ordered pair, and (a, b)R for rectangular coordinate
ordered pair.
Conversion Rules
Let (x, y)R and (r, θ)P be the rectangular and polar
coordinates of the same point P, then
x = r*cos(θ) y
(r, θ) = (x, y)
R P
y = r*sin(θ) P
r = √ x2 + y2
For θ we have r
y = r*sin(θ)
tan(θ) = y/x
cos(θ) = x/√x2 + y2 θ x
x = r*cos(θ)
The rectangular and polar
coordinates relations
16. Polar Coordinates
If needed, we write (a, b)P for a polar coordinate
ordered pair, and (a, b)R for rectangular coordinate
ordered pair.
Conversion Rules
Let (x, y)R and (r, θ)P be the rectangular and polar
coordinates of the same point P, then
x = r*cos(θ) y
(r, θ) = (x, y)
R P
y = r*sin(θ) P
r = √ x2 + y2
For θ we have r
y = r*sin(θ)
tan(θ) = y/x
cos(θ) = x/√x2 + y2 or using θ x
inverse trig. functions that O x = r*cos(θ)
θ = tan–1(y/x) The rectangular and polar
θ = cos–1 (x/√x2 + y2) coordinates relations
17. Polar Coordinates
Example A. a. Plot the following polar coordinates
A(4, 60o)P , B(5, 0o)P, C(4, –45o)P, D(–4, 3π/4 rad)P.
Find their corresponding rectangular coordinates.
18. Polar Coordinates
Example A. a. Plot the following polar coordinates
A(4, 60o)P , B(5, 0o)P, C(4, –45o)P, D(–4, 3π/4 rad)P.
Find their corresponding rectangular coordinates.
For A(4, 60o)P y
x
x = r*cos(θ) r2 = x2 + y2
y = r*sin(θ) tan(θ) = y/x
19. Polar Coordinates
Example A. a. Plot the following polar coordinates
A(4, 60o)P , B(5, 0o)P, C(4, –45o)P, D(–4, 3π/4 rad)P.
Find their corresponding rectangular coordinates.
For A(4, 60o)P y A(4, 60 ) o
P
4
60o x
x = r*cos(θ) r2 = x2 + y2
y = r*sin(θ) tan(θ) = y/x
20. Polar Coordinates
Example A. a. Plot the following polar coordinates
A(4, 60o)P , B(5, 0o)P, C(4, –45o)P, D(–4, 3π/4 rad)P.
Find their corresponding rectangular coordinates.
For A(4, 60o)P y A(4, 60 ) o
P
(x, y)R = (4*cos(60), 4*sin(60)),
4
60o x
x = r*cos(θ) r2 = x2 + y2
y = r*sin(θ) tan(θ) = y/x
21. Polar Coordinates
Example A. a. Plot the following polar coordinates
A(4, 60o)P , B(5, 0o)P, C(4, –45o)P, D(–4, 3π/4 rad)P.
Find their corresponding rectangular coordinates.
For A(4, 60o)P y A(4, 60 ) o
P
(x, y)R = (4*cos(60), 4*sin(60)),
4
= (2, 2√3)
60o x
x = r*cos(θ) r2 = x2 + y2
y = r*sin(θ) tan(θ) = y/x
22. Polar Coordinates
Example A. a. Plot the following polar coordinates
A(4, 60o)P , B(5, 0o)P, C(4, –45o)P, D(–4, 3π/4 rad)P.
Find their corresponding rectangular coordinates.
For A(4, 60o)P y A(4, 60 ) o
P
(x, y)R = (4*cos(60), 4*sin(60)),
4
= (2, 2√3)
for B(5, 0o)P, (x, y) = (5, 0), 60 o
x
B(5, 0)P
x = r*cos(θ) r2 = x2 + y2
y = r*sin(θ) tan(θ) = y/x
23. Polar Coordinates
Example A. a. Plot the following polar coordinates
A(4, 60o)P , B(5, 0o)P, C(4, –45o)P, D(–4, 3π/4 rad)P.
Find their corresponding rectangular coordinates.
For A(4, 60o)P y A(4, 60 ) o
P
(x, y)R = (4*cos(60), 4*sin(60)),
4
= (2, 2√3)
for B(5, 0o)P, (x, y) = (5, 0), 60 o
x
–45 o
B(5, 0)
for C and D, P
4
C
x = r*cos(θ) r2 = x2 + y2
y = r*sin(θ) tan(θ) = y/x
24. Polar Coordinates
Example A. a. Plot the following polar coordinates
A(4, 60o)P , B(5, 0o)P, C(4, –45o)P, D(–4, 3π/4 rad)P.
Find their corresponding rectangular coordinates.
For A(4, 60o)P y A(4, 60 ) o
P
(x, y)R = (4*cos(60), 4*sin(60)), 3π/4
4
= (2, 2√3)
for B(5, 0o)P, (x, y) = (5, 0), 60 o
x
–45 o
B(5, 0)
for C and D, P
4 C
&
D
x = r*cos(θ) r2 = x2 + y2
y = r*sin(θ) tan(θ) = y/x
25. Polar Coordinates
Example A. a. Plot the following polar coordinates
A(4, 60o)P , B(5, 0o)P, C(4, –45o)P, D(–4, 3π/4 rad)P.
Find their corresponding rectangular coordinates.
For A(4, 60o)P D
y A(4, 60 ) o
P
(x, y)R = (4*cos(60), 4*sin(60)), 3π/4
4
= (2, 2√3)
for B(5, 0o)P, (x, y) = (5, 0), 60 o
x
–45 o
B(5, 0)
for C and D, P
4
C
C(4, –45o)P
= D(–4, 3π/4 rad)P
x = r*cos(θ) r2 = x2 + y2
y = r*sin(θ) tan(θ) = y/x
26. Polar Coordinates
Example A. a. Plot the following polar coordinates
A(4, 60o)P , B(5, 0o)P, C(4, –45o)P, D(–4, 3π/4 rad)P.
Find their corresponding rectangular coordinates.
For A(4, 60o)P D
y A(4, 60 ) o
P
(x, y)R = (4*cos(60), 4*sin(60)), 3π/4
4
= (2, 2√3)
for B(5, 0o)P, (x, y) = (5, 0), 60 o
x
–45 o
B(5, 0)
for C and D, P
(x, y)R = (4cos(–45), 4sin(–45)) 4
C
= (–4cos(3π/4), –4sin(3π/4)) C(4, –45 ) o
P
= D(–4, 3π/4 rad)P
x = r*cos(θ) r2 = x2 + y2
y = r*sin(θ) tan(θ) = y/x
27. Polar Coordinates
Example A. a. Plot the following polar coordinates
A(4, 60o)P , B(5, 0o)P, C(4, –45o)P, D(–4, 3π/4 rad)P.
Find their corresponding rectangular coordinates.
For A(4, 60o)P D
y A(4, 60 ) o
P
(x, y)R = (4*cos(60), 4*sin(60)), 3π/4
4
= (2, 2√3)
for B(5, 0o)P, (x, y) = (5, 0), 60 o
x
–45 o
B(5, 0)
for C and D, P
(x, y)R = (4cos(–45), 4sin(–45)) 4
C
= (–4cos(3π/4), –4sin(3π/4)) C(4, –45 ) o
P
= (2√2, –2√2) = D(–4, 3π/4 rad) P
x = r*cos(θ) r2 = x2 + y2
y = r*sin(θ) tan(θ) = y/x
28. Polar Coordinates
Example A. a. Plot the following polar coordinates
A(4, 60o)P , B(5, 0o)P, C(4, –45o)P, D(–4, 3π/4 rad)P.
Find their corresponding rectangular coordinates.
For A(4, 60o)P D
y A(4, 60 ) o
P
(x, y)R = (4*cos(60), 4*sin(60)), 3π/4
4
= (2, 2√3)
for B(5, 0o)P, (x, y) = (5, 0), 60 o
x
–45 o
B(5, 0)
for C and D, P
(x, y)R = (4cos(–45), 4sin(–45)) 4
C
= (–4cos(3π/4), –4sin(3π/4)) C(4, –45 ) o
P
= (2√2, –2√2) = D(–4, 3π/4 rad) P
Converting rectangular positions
into polar coordinates requires x = r*cos(θ) r = x + y
2 2 2
y = r*sin(θ) tan(θ) = y/x
more care.
29. Polar Coordinates
b. Find a polar coordinate then list all possible polar
coordinates for each of the following points
(with r > 0): E(–4, 3)R, F(3, –2)R, and G(–3, –1)R.
30. Polar Coordinates
b. Find a polar coordinate then list all possible polar
coordinates for each of the following points
(with r > 0): E(–4, 3)R, F(3, –2)R, and G(–3, –1)R.
y
E(–4, 3)
x
31. Polar Coordinates
b. Find a polar coordinate then list all possible polar
coordinates for each of the following points
(with r > 0): E(–4, 3)R, F(3, –2)R, and G(–3, –1)R.
We have the distance formula r = √x2 + y2,
y
E(–4, 3)
x
32. Polar Coordinates
b. Find a polar coordinate then list all possible polar
coordinates for each of the following points
(with r > 0): E(–4, 3)R, F(3, –2)R, and G(–3, –1)R.
We have the distance formula r = √x2 + y2,
hence for E, r = √16 + 9 = 5. y
E(–4, 3)
r=5
x
33. Polar Coordinates
b. Find a polar coordinate then list all possible polar
coordinates for each of the following points
(with r > 0): E(–4, 3)R, F(3, –2)R, and G(–3, –1)R.
We have the distance formula r = √x2 + y2,
hence for E, r = √16 + 9 = 5. y
There is no single formula that
would give θ. E(–4, 3)
θ
r=5
x
34. Polar Coordinates
b. Find a polar coordinate then list all possible polar
coordinates for each of the following points
(with r > 0): E(–4, 3)R, F(3, –2)R, and G(–3, –1)R.
We have the distance formula r = √x2 + y2,
hence for E, r = √16 + 9 = 5. y
There is no single formula that
would give θ. This is because θ has E(–4, 3) θ
to be expressed via the inverse r=5
x
trig–functions hence the position of
E dictates which inverse function
would be easier to use to extract θ.
35. Polar Coordinates
b. Find a polar coordinate then list all possible polar
coordinates for each of the following points
(with r > 0): E(–4, 3)R, F(3, –2)R, and G(–3, –1)R.
We have the distance formula r = √x2 + y2,
hence for E, r = √16 + 9 = 5. y
There is no single formula that
would give θ. This is because θ has E(–4, 3) θ
to be expressed via the inverse r=5
x
trig–functions hence the position of
E dictates which inverse function
would be easier to use to extract θ. Since E is in the
2nd quadrant, the angle θ may be recovered by the
cosine inverse function (why?).
36. Polar Coordinates
b. Find a polar coordinate then list all possible polar
coordinates for each of the following points
(with r > 0): E(–4, 3)R, F(3, –2)R, and G(–3, –1)R.
We have the distance formula r = √x2 + y2,
hence for E, r = √16 + 9 = 5. y
There is no single formula that
would give θ. This is because θ has E(–4, 3) θ
to be expressed via the inverse r=5
x
trig–functions hence the position of
E dictates which inverse function
would be easier to use to extract θ. Since E is the 2nd
quadrant, the angle θ may be recovered by the cosine
inverse function (why?). So θ = cos–1(–4/5) ≈ 143o
37. Polar Coordinates
b. Find a polar coordinate then list all possible polar
coordinates for each of the following points
(with r > 0): E(–4, 3)R, F(3, –2)R, and G(–3, –1)R.
We have the distance formula r = √x2 + y2,
hence for E, r = √16 + 9 = 5. y
There is no single formula that
would give θ. This is because θ has E(–4, 3) θ
to be expressed via the inverse r=5
x
trig–functions hence the position of
E dictates which inverse function
would be easier to use to extract θ. Since E is the 2nd
quadrant, the angle θ may be recovered by the cosine
inverse function (why?). So θ = cos–1(–4/5) ≈ 143o
or that E(–4, 3)R ≈ (5, 143o)P
38. Polar Coordinates
b. Find a polar coordinate then list all possible polar
coordinates for each of the following points
(with r > 0): E(–4, 3)R, F(3, –2)R, and G(–3, –1)R.
We have the distance formula r = √x2 + y2,
hence for E, r = √16 + 9 = 5. y
There is no single formula that
would give θ. This is because θ has E(–4, 3) θ
to be expressed via the inverse r=5
x
trig–functions hence the position of
E dictates which inverse function
would be easier to use to extract θ. Since E is the 2nd
quadrant, the angle θ may be recovered by the cosine
inverse function (why?). So θ = cos–1(–4/5) ≈ 143o
or that E(–4, 3)R ≈ (5, 143o)P = (5, 143o±n*360o)P
41. Polar Coordinates
For F(3, –2)R, r = √9 + 4 = √13. y
Since F is in the 4th quadrant, the x
angle θ may be recovered by the θ
sine inverse or the tangent inverse r=√13
function. F(3, –2,)
42. Polar Coordinates
For F(3, –2)R, r = √9 + 4 = √13. y
Since F is the 4th quadrant, the x
angle θ may be recovered by the θ
sine inverse or the tangent inverse r=√13
function. The tangent inverse has F(3, –2,)
the advantage of obtaining the answer directly from
the x and y coordinates.
43. Polar Coordinates
For F(3, –2)R, r = √9 + 4 = √13. y
Since F is the 4th quadrant, the x
angle θ may be recovered by the θ
sine inverse or the tangent inverse r=√13
function. The tangent inverse has F(3, –2,)
the advantage of obtaining the answer directly from
the x and y coordinates. So θ = tan–1(–2/3)
≈ –0.588rad and that F(3, –2)R ≈ (√13, –0.588rad)P
44. Polar Coordinates
For F(3, –2)R, r = √9 + 4 = √13. y
Since F is the 4th quadrant, the x
angle θ may be recovered by the θ
sine inverse or the tangent inverse r=√13
function. The tangent inverse has F(3, –2,)
the advantage of obtaining the answer directly from
the x and y coordinates. So θ = tan–1(–2/3)
≈ –0.588rad and that F(3, –2)R ≈ (√13, –0.588rad)P
= (√13, –0.588rad ± 2nπ)P
45. Polar Coordinates
For F(3, –2)R, r = √9 + 4 = √13. y
Since F is the 4th quadrant, the x
angle θ may be recovered by the θ
sine inverse or the tangent inverse r=√13
function. The tangent inverse has F(3, –2,)
the advantage of obtaining the answer directly from
the x and y coordinates. So θ = tan–1(–2/3)
≈ –0.588rad and that F(3, –2)R ≈ (√13, –0.588rad)P
= (√13, –0.588rad ± 2nπ)P
y
For G(–3, –1)R, r = √9 + 1 = √10.
x
r=√10
G(–3, –1)
46. Polar Coordinates
For F(3, –2)R, r = √9 + 4 = √13. y
Since F is the 4th quadrant, the x
angle θ may be recovered by the θ
sine inverse or the tangent inverse r=√13
function. The tangent inverse has F(3, –2,)
the advantage of obtaining the answer directly from
the x and y coordinates. So θ = tan–1(–2/3)
≈ –0.588rad and that F(3, –2)R ≈ (√13, –0.588rad)P
= (√13, –0.588rad ± 2nπ)P
y
For G(–3, –1)R, r = √9 + 1 = √10.
G is the 3rd quadrant. Hence θ can’t x
be obtained directly via the inverse– r=√10
trig functions. G(–3, –1)
47. Polar Coordinates
For F(3, –2)R, r = √9 + 4 = √13. y
Since F is the 4th quadrant, the x
angle θ may be recovered by the θ
sine inverse or the tangent inverse r=√13
function. The tangent inverse has F(3, –2,)
the advantage of obtaining the answer directly from
the x and y coordinates. So θ = tan–1(–2/3)
≈ –0.588rad and that F(3, –2)R ≈ (√13, –0.588rad)P
= (√13, –0.588rad ± 2nπ)P
y
For G(–3, –1)R, r = √9 + 1 = √10.
G is the 3rd quadrant. Hence θ can’t x
A
be obtained directly via the inverse– r=√10
trig functions. We will find the angle G(–3, –1)
A as shown first, then θ = A + π.
48. Polar Coordinates
y
Again, using tangent inverse x
A = tan–1(1/3) ≈ 18.4o A
r=√10
G(–3, –1)
49. Polar Coordinates
y
Again, using tangent inverse θ
x
A = tan–1(1/3) ≈ 18.3o so A
θ = 180 + 18.3o = 198.3o r=√10
G(–3, –1)
50. Polar Coordinates
y
Again, using tangent inverse θ
x
A = tan–1(1/3) ≈ 18.4o so A
θ = 180 + 18.4o = 198.4o or r=√10
G ≈ (√10, 198.4o ± n x 360o)P G(–3, –1)
51. Polar Coordinates
y
Again, using tangent inverse θ
x
A = tan–1(1/3) ≈ 18.3o so A
θ = 180 + 18.3o = 198.3o or r=√10
G ≈ (√10, 198.3o ± n x 360o)P G(–3, –1)
Polar Equations
52. Polar Coordinates
y
Again, using tangent inverse θ
x
A = tan–1(1/3) ≈ 18.3o so A
θ = 180 + 18.3o = 198.3o or r=√10
G ≈ (√10, 198.3o ± n x 360o)P G(–3, –1)
Polar Equations
A polar equation is a description of the relation
between points using their distances and directions.
53. Polar Coordinates
y
Again, using tangent inverse θ
x
A = tan–1(1/3) ≈ 18.3o so A
θ = 180 + 18.3o = 198.3o or r=√10
G ≈ (√10, 198.3o ± n x 360o)P G(–3, –1)
Polar Equations
A polar equation is a description of the relation
between points using their distances and directions. In
symbols, polar equations look like our old equations
except that x and y are replaced with r and θ.
54. Polar Coordinates
y
Again, using tangent inverse θ
x
A = tan–1(1/3) ≈ 18.3o so A
θ = 180 + 18.3o = 198.3o or r=√10
G ≈ (√10, 198.3o ± n x 360o)P G(–3, –1)
Polar Equations
A polar equation is a description of relation between
points using their distances and directions. In symbols,
polar equations look like our old equations except that
x and y are replaced with r and θ. However, the
geometry described respectively by the symbols are
completely different.
55. Polar Coordinates
y
Again, using tangent inverse θ
x
A = tan–1(1/3) ≈ 18.3o so A
θ = 180 + 18.3o = 198.3o or r=√10
G ≈ (√10, 198.3o ± n x 360o)P G(–3, –1)
Polar Equations
A polar equation is a description of relation between
points using their distances and directions. In symbols,
polar equations look like our old equations except that
x and y are replaced with r and θ. However, the
geometry described respectively by the symbols are
completely different. We use the x = r*cos(θ)
conversion rules to translate y = r*sin(θ)
equations between the two systems. r = √x2 + y2
tan(θ) = y/x
56. Polar Coordinates
y
Again, using tangent inverse θ
x
A = tan–1(1/3) ≈ 18.3o so A
θ = 180 + 18.3o = 198.3o or r=√10
G ≈ (√10, 198.3o ± n x 360o)P G(–3, –1)
Polar Equations
A polar equation is a description of relation between
points using their distances and directions. In symbols,
polar equations look like our old equations except that
x and y are replaced with r and θ. However, the
geometry described respectively by the symbols are
completely different. We use the x = r*cos(θ)
conversion rules to translate y = r*sin(θ)
equations between the two systems. r = √x2 + y2
Let’s start with some basic equations. tan(θ) = y/x
57. Polar Coordinates
Equations in x and y are called rectangular equations
and equations in r and θ are called polar equations.
58. Polar Coordinates
Equations in x and y are called rectangular equations
and equations in r and θ are called polar equations.
Example B. Convert each of the following rectangular
equations into the corresponding polar form.
Write the answer in the r = f(θ) form and interpret its
geometric significances in terms of distances and
directions.
59. Polar Coordinates
Equations in x and y are called rectangular equations
and equations in r and θ are called polar equations.
Example B. Convert each of the following rectangular
equations into the corresponding polar form.
Write the answer in the r = f(θ) form and interpret its
geometric significances in terms of distances and
directions.
a. x = k
x
x=k
60. Polar Coordinates
Equations in x and y are called rectangular equations
and equations in r and θ are called polar equations.
Example B. Convert each of the following rectangular
equations into the corresponding polar form.
Write the answer in the r = f(θ) form and interpret its
geometric significances in terms of distances and
directions.
a. x = k
Replacing x with r*cos(θ)
we get that r*cos(θ) = k
x
x=k
61. Polar Coordinates
Equations in x and y are called rectangular equations
and equations in r and θ are called polar equations.
Example B. Convert each of the following rectangular
equations into the corresponding polar form.
Write the answer in the r = f(θ) form and interpret its
geometric significances in terms of distances and
directions.
a. x = k
Replacing x with r*cos(θ)
we get that r*cos(θ) = k
or that r = k*sec(θ) = f(θ). x
x=k
62. Polar Coordinates
Equations in x and y are called rectangular equations
and equations in r and θ are called polar equations.
Example B. Convert each of the following rectangular
equations into the corresponding polar form.
Write the answer in the r = f(θ) form and interpret its
geometric significances in terms of distances and
directions.
a. x = k
Replacing x with r*cos(θ)
we get that r*cos(θ) = k
or that r = k*sec(θ) = f(θ). x
In the picture, r = k*sec(θ) gives
the basic trig. relation of points
on the vertical line x = k as
x=k
63. Polar Coordinates
Equations in x and y are called rectangular equations
and equations in r and θ are called polar equations.
Example B. Convert each of the following rectangular
equations into the corresponding polar form.
Write the answer in the r = f(θ) form and interpret its
geometric significances in terms of distances and
directions.
a. x = k (k, y)
Replacing x with r*cos(θ)
we get that r*cos(θ) = k
or that r = k*sec(θ) = f(θ). θ x
In picture, r = k*sec(θ) gives the k
basic trig. relation of points on
the vertical line x = k as shown.
x=k
64. Polar Coordinates
Equations in x and y are called rectangular equations
and equations in r and θ are called polar equations.
Example B. Convert each of the following rectangular
equations into the corresponding polar form.
Write the answer in the r = f(θ) form and interpret its
geometric significances in terms of distances and
directions.
a. x = k (k, y)
Replacing x with r*cos(θ)
we get that r*cos(θ) = k r = k*sec(θ )
or that r = k*sec(θ) = f(θ). θ x
In picture, r = k*sec(θ) gives the k
basic trig. relation of points on
the vertical line x = k as shown.
x=k
66. Polar Coordinates
b. y = x y=x
For y = x, replace x = r*cos(θ)
and y = r*sin(θ), we get that
r*sin(θ) = r*cos(θ), x
67. Polar Coordinates
b. y = x y=x
For y = x, replace x = r*cos(θ)
and y = r*sin(θ), we get that
r*sin(θ) = r*cos(θ), assuming r ≠ 0, x
we have sin(θ) = cos(θ),
68. Polar Coordinates
b. y = x y=x
For y = x, replace x = r*cos(θ)
and y = r*sin(θ), we get that
r*sin(θ) = r*cos(θ), assuming r ≠ 0, x
we have sin(θ) = cos(θ),
dividing by cos(θ) we have that
tan(θ) = 1
69. Polar Coordinates
b. y = x y=x
For y = x, replace x = r*cos(θ)
and y = r*sin(θ), we get that
θ = π/4.
r*sin(θ) = r*cos(θ), assuming r ≠ 0, x
we have sin(θ) = cos(θ),
dividing by cos(θ) we have that
tan(θ) = 1 or that θ
= π/4 ± nπ.
70. Polar Coordinates
b. y = x y=x
For y = x, replace x = r*cos(θ)
and y = r*sin(θ), we get that
θ = π/4.
r*sin(θ) = r*cos(θ), assuming r ≠ 0, x
we have sin(θ) = cos(θ),
dividing by cos(θ) we have that
tan(θ) = 1 or that θ
The equation θ = π/4 is a polar constant equation.
= π/4 ± nπ.
71. Polar Coordinates
b. y = x y=x
For y = x, replace x = r*cos(θ)
and y = r*sin(θ), we get that
θ = π/4.
r*sin(θ) = r*cos(θ), assuming r ≠ 0, x
we have sin(θ) = cos(θ),
dividing by cos(θ) we have that
tan(θ) = 1 or that θ
The equation θ = π/4 is a polar constant equation.
= π/4 ± nπ.
Since the variable r is missing, r can be of any value.
72. Polar Coordinates
b. y = x y=x
For y = x, replace x = r*cos(θ)
and y = r*sin(θ), we get that
θ = π/4.
r*sin(θ) = r*cos(θ), assuming r ≠ 0, x
we have sin(θ) = cos(θ),
dividing by cos(θ) we have that
tan(θ) = 1 or that θ
The equation θ = π/4 is a polar constant equation.
= π/4 ± nπ.
Since the variable r is missing, r can be of any value.
Geometrically, it says that the diagonal line y = x
consists of those points whose polar angles θ = π/4.
73. Polar Coordinates
b. y = x y=x
For y = x, replace x = r*cos(θ)
and y = r*sin(θ), we get that
θ = π/4.
r*sin(θ) = r*cos(θ), assuming r ≠ 0, x
we have sin(θ) = cos(θ),
dividing by cos(θ) we have that
tan(θ) = 1 or that θ
The equation θ = π/4 is a polar constant equation.
= π/4 ± nπ.
Since the variable r is missing, r can be of any value.
Geometrically, it says that the diagonal line y = x
consists of those points whose polar angles θ = π/4.
Note that there are infinitely many non–equivalent
polar equations that define the same set of diagonal
points.
74. Polar Coordinates
b. y = x y=x
For y = x, replace x = r*cos(θ)
and y = r*sin(θ), we get that
θ = π/4.
r*sin(θ) = r*cos(θ), assuming r ≠ 0, x
we have sin(θ) = cos(θ),
dividing by cos(θ) we have that
tan(θ) = 1 or that θ
The equation θ = π/4 is a polar constant equation.
= π/4 ± nπ.
Since the variable r is missing, r can be of any value.
Geometrically, it says that the diagonal line y = x
consists of those points whose polar angles θ = π/4.
Note that there are infinitely many non–equivalent
polar equations that define the same set of diagonal
points. Specifically that θ = 5π/4, θ = 9π/4, ..,
θ = –3π/4, θ = –7π/4, .. all give y = x.
76. Polar Coordinates
The non–uniqueness of the polar form is the major
difference between the rectangular & polar systems.
Steps employed and solutions obtained in the
rectangular x&y equations, when applied to the r&θ
polar equations, have to be reinterpreted in light of the
polar geometry.
77. Polar Coordinates
The non–uniqueness of the polar form is the major
difference between the rectangular & polar systems.
Steps employed and solutions obtained in the
rectangular x&y equations, when applied to the r&θ
polar equations, have to be reinterpreted in light of the
polar geometry.
Example C. Interpret and draw the graph of each of
the following polar equations. Convert each equation
into a corresponding rectangular form.
a. r = k
78. Polar Coordinates
The non–uniqueness of the polar form is the major
difference between the rectangular & polar systems.
Steps employed and solutions obtained in the
rectangular x&y equations, when applied to the r&θ
polar equations, have to be reinterpreted in light of the
polar geometry.
Example C. Interpret and draw the graph of each of
the following polar equations. Convert each equation
into a corresponding rectangular form.
a. r = k
The polar equation states that the
distance r, from the origin to our points,
is a constant k.
79. Polar Coordinates
The non–uniqueness of the polar form is the major
difference between the rectangular & polar systems.
Steps employed and solutions obtained in the
rectangular x&y equations, when applied to the r&θ
polar equations, have to be reinterpreted in light of the
polar geometry.
Example C. Interpret and draw the graph of each of
the following polar equations. Convert each equation
into a corresponding rectangular form.
a. r = k
The polar equation states that the r=k x
distance r, from the origin to our points,
is a constant k. This is the circle of
radius k, centered at (0, 0).
80. Polar Coordinates
Set r = √x2 + y2 = k we have that
x2 + y2 = k2 in the rectangular form.
r=k x
81. Polar Coordinates
Set r = √x2 + y2 = k we have that
x2 + y2 = k2 in the rectangular form.
r=k x
b. r = θ
82. Polar Coordinates
Set r = √x2 + y2 = k we have that
x2 + y2 = k2 in the rectangular form.
r=k x
b. r = θ
Let θ > 0 (in radian), the polar equation
states that the distance r is of the same
as θ.
83. Polar Coordinates
Set r = √x2 + y2 = k we have that
x2 + y2 = k2 in the rectangular form.
r=k x
b. r = θ
Let θ > 0 (in radian), the polar equation
states that the distance r is of the same
as θ. Hence starting at (0, 0)P, as
θ increases, r increases, so the points
are circling outward from the
origin at a steady or linear rate.
84. Polar Coordinates
Set r = √x2 + y2 = k we have that
x2 + y2 = k2 in the rectangular form.
r=k x
b. r = θ
Let θ > 0 (in radian), the polar equation
states that the distance r is of the same
as θ. Hence starting at (0, 0)P, as
θ increases, r increases, so the points x
are circling outward from the
origin at a steady or linear rate. r=θ
In general, the graph of r = f(θ) where
f(θ) is an increasing or decreasing
function is called a spiral.
85. Polar Coordinates
Set r = √x2 + y2 = k we have that
x2 + y2 = k2 in the rectangular form.
r=k x
b. r = θ
Let θ > 0 (in radian), the polar equation
states that the distance r is of the same
as θ. Hence starting at (0, 0)P, as
θ increases, r increases, so the points x
are circling outward from the
origin at a steady or linear rate. r=θ
In general, the graph of r = f(θ) where Archimedean spirals
f(θ) is an increasing or decreasing
function is called a spiral. A uniformly
banded spiral such as this one is called
an Archimedean spiral.
87. Polar Coordinates
We will use cosine inverse
function to express θ in x&y, i.e.
θ = cos–1(x/r) = cos–1(x/√x2 + y2 ). x
x
r=θ
88. Polar Coordinates
We will use cosine inverse
function to express θ in x&y, i.e.
θ = cos–1(x/r) = cos–1(x/√x2 + y2 ). x
We have the equation that
cos–1(x/√x2 + y2) = √x2 + y2 ( = r ) x
r=θ
89. Polar Coordinates
We will use cosine inverse
function to express θ in x&y, i.e.
θ = cos–1(x/r) = cos–1(x/√x2 + y2 ). x
We have the equation that
cos–1(x/√x2 + y2) = √x2 + y2 ( = r ) x
This rectangular equation only r=θ
gives the part of the spiral where
0 < √x2 + y2 ≤ π (why?)
90. Polar Coordinates
We will use cosine inverse
function to express θ in x&y, i.e.
θ = cos–1(x/r) = cos–1(x/√x2 + y2 ). x
We have the equation that
cos–1(x/√x2 + y2) = √x2 + y2 ( = r ) x
This rectangular equation only r=θ
gives the part of the spiral where
0 < √x2 + y2 ≤ π (why?)
x
cos–1(x/√x2 + y2) = √x2 + y2
The “Lost in Translation”
from the polar to the
rectangular equation
91. Polar Coordinates
We will use cosine inverse
function to express θ in x&y, i.e.
θ = cos–1(x/r) = cos–1(x/√x2 + y2 ). x
We have the equation that
cos–1(x/√x2 + y2) = √x2 + y2 ( = r ) x
This rectangular equation only r=θ
gives the part of the spiral where
0 < √x2 + y2 ≤ π (why?)
For other segments of the
spirals, we add nπ with n = 1,2,..
x
cos–1(x/√x2 + y2) = √x2 + y2
The “Lost in Translation”
from the polar to the
rectangular equation
92. Polar Coordinates
We will use cosine inverse
function to express θ in x&y, i.e.
θ = cos–1(x/r) = cos–1(x/√x2 + y2 ). x
We have the equation that
cos–1(x/√x2 + y2) = √x2 + y2 ( = r ) x
This rectangular equation only r=θ
gives the part of the spiral where
0 < √x2 + y2 ≤ π (why?)
For other parts of the spirals,
we add nπ to θ with n = 1,2,.. to
obtain more distant segments, so x
cos–1(x/√x2 + y2) + nπ = √x2 + y2. cos–1(x/√x2 + y2) = √x2 + y2
The “Lost in Translation”
from the polar to the
rectangular equation
93. Polar Coordinates
We will use cosine inverse
function to express θ in x&y, i.e.
θ = cos–1(x/r) = cos–1(x/√x2 + y2 ). x
We have the equation that
cos–1(x/√x2 + y2) = √x2 + y2 ( = r ) x
This rectangular equation only r=θ
gives the part of the spiral where
0 < √x2 + y2 ≤ π (why?)
For other parts of the spirals,
we add nπ to θ with n = 1,2,.. to
obtain more distant segments, so x
cos–1(x/√x2 + y2) + nπ = √x2 + y2. cos–1(x/√x2 + y2) = √x2 + y2
The “Lost in Translation”
This shows the advantages of the from the polar to the
polar system in certain settings. rectangular equation