The programme explains the concept of trigonometry.It also attempts to explain various parts of a right angled triangle -hypotenuse,adjacent side and opposite sides.It also gives the explanation of trigonometric ratios-sine,cosine and tangents.
Arithmetic progression
For class 10.
In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant
This PPT will clarify your all doubts in Arithmetic Progression.
Please download this PPT and if any doubt according to this PPT, please comment , then i will try to solve your problem.
Thank you :)
A plane figure with three sides and three angles is called a triangle. We will learn the different types of triangles based on varying side lengths and angle measurements. After this session you can very easily tell the difference between all types of triangles and know the mathematics involved in it.
Did you know, two different triangles of different sizes can be similar to each other based on the ratio of their sides ?
Here you will learn the following:
1) Criteria’s for similarity
2) Scale factor
3) Congruency
If the corresponding sides of a triangle is twice than that of another triangle, will the area be also doubled??
Watch this session to learn about the effects that can be seen in areas of two similar triangles in just 10 minutes.
Basic Proportionality Theorem is one of the important topics of a Triangle that deals with the study of the proportion of the two sides of a triangle. So, watch this session and learn about the Theorem and its proof.
Pythagoras theorem is a fundamental relation in Euclidean geometry among the three sides of a right triangle.
In this session, you will learn this very important theorem and learn to prove its statement with its proof in a geometric way.
Arithmetic progression
For class 10.
In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant
This PPT will clarify your all doubts in Arithmetic Progression.
Please download this PPT and if any doubt according to this PPT, please comment , then i will try to solve your problem.
Thank you :)
A plane figure with three sides and three angles is called a triangle. We will learn the different types of triangles based on varying side lengths and angle measurements. After this session you can very easily tell the difference between all types of triangles and know the mathematics involved in it.
Did you know, two different triangles of different sizes can be similar to each other based on the ratio of their sides ?
Here you will learn the following:
1) Criteria’s for similarity
2) Scale factor
3) Congruency
If the corresponding sides of a triangle is twice than that of another triangle, will the area be also doubled??
Watch this session to learn about the effects that can be seen in areas of two similar triangles in just 10 minutes.
Basic Proportionality Theorem is one of the important topics of a Triangle that deals with the study of the proportion of the two sides of a triangle. So, watch this session and learn about the Theorem and its proof.
Pythagoras theorem is a fundamental relation in Euclidean geometry among the three sides of a right triangle.
In this session, you will learn this very important theorem and learn to prove its statement with its proof in a geometric way.
Surface area of a cuboid and a cube,cylinder,cone,sphere,volume of cuboid,cyl...kamal brar
surface area of a cuboid and a cube,surface area of a right circular cylinder,surface area of right circular cone,surface area of a sphere,volume of cuboid,volume of cylinder,volume of right circular cone and volume of sphere.powerpoint presentation
The power point explains the formation of an angle,different types of angles,complementary and supplementary angles and vertical angles with suitable examples.
Surface area of a cuboid and a cube,cylinder,cone,sphere,volume of cuboid,cyl...kamal brar
surface area of a cuboid and a cube,surface area of a right circular cylinder,surface area of right circular cone,surface area of a sphere,volume of cuboid,volume of cylinder,volume of right circular cone and volume of sphere.powerpoint presentation
The power point explains the formation of an angle,different types of angles,complementary and supplementary angles and vertical angles with suitable examples.
Inductive method:a psychological method of developing formulas and principles
Deductive method:A speedy method of deduction and application.
best method is to develop formuias and then apply in examples therefore -inducto -deductive method
Review of Trigonometry for Calculus “Trigon” =triangle +“metry”=measurement =...KyungKoh2
Review of Trigonometry for Calculus “Trigon” =triangle +“metry”=measurement =Trigonometry so Trigonometry got its name as the science of measuring triangles.
The derivation of the spherical gradient from an intuitive perspective. We begin from Cartesian coordinates and work our way through to spherical coordinates.
The power point explains the concept of quadrilateral.It also helps us to understand important theorem " the sum of all the angles in a quadrilateral is 180 degrees".
The power point explains about the life history and contribution of Pythagoras.It also helps us to understand the development of the Pythagoras formula.It also attempts to solve few problems based on Pythagoras.
The power point explains different concepts of the parts of circle.It helps us to understand the concepts of secants and tangents also with the help of examples.
The power point explains the concept of pairs of angles and transversal with the help of examples.It also helps us to understand the concepts of complementary and supplementary angles.
The power point explains the concept of congruence in VII th standard .It explains the congruence of angles,vertices, triangles,quadrilaterals,and circle.
The power point explains the concept of simple interest and compound interest.It also explains the development of the formula.It is made for VIIIth standard S.S.C text book.
The power point is based on the concept attainment model of teaching mathematics.It explains the parts of circle-Radius ,diameter,center,relationship between radius and diameter,chord,properties of radius,chord and diameter,types of arcs and the exterior and interior of the circle.
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
2. Trigonometry is the
branch of mathematics
which deals with
triangles, particularly
triangles in a plane
where one angle of the
triangle is 90 degrees.
.
Trigonometry
2
3. Trigonometry specifically deals with
the relationships between the
sides and the angles of triangles.
----------------------------------------
--------
3
4. In trigonometry, the ratio we are
talking about is the comparison of
the sides of a RIGHT ANGLED
TRIANGLE. Two things MUST BE understood:
1. This is the hypotenuse.
.
2. This is 90°
5. Now that we agree about the hypotenuse
and right angle, there are only 4 things left;
the 2 other angles and the 2 other sides.
A
.
Opposite side
Adjacent side
Hypotenuse
7. One more thing…
θ this is the symbol for an unknown angle
measure.
It’s name is ‘Theta’.
8. Trigonometric Ratios
Name
“say” Sine Cosine tangent
Abbreviation
Sin Cos Tan
Ratio of an
angle
measure
Sinθ = opposite side
hypotenuse
cosθ = adjacent side
hypotenuse
tanθ =opposite side
adjacent side
9. One more
time…
Here are the
ratios:
sinθ = opposite side
hypotenuse
cosθ = adjacent side
hypotenuse
tanθ =opposite side
adjacent side
10. Trigonometric Identities
A trigonometric equation is an equation that involves
at least one trigonometric function of a variable. The
equation is a trigonometric identity if it is true for all
values of the variable for which both sides of the
equation are defined.
Prove that tan
sin
cos
.
y
x
y
r
x
r
y
r
r
x
y
x
L.S. = R.S.
5.4.2
Recall the basic
trig identities:
sin
y
r
cos
x
r
tan
y
x
12. Trigonometric Identities [cont’d]
sinx x sinx = sin2x
cos
1
cos
cos2
cos
1
cos
cos2 1
cos
sinA cos A2
sin2 A 2sinAcos A cos2 A
12sinAcosA
cosA
sinA
1
sinA
cosA
sinA
sinA
1
= cosA
5.4.4
13. Simplifying Trigonometric Expressions
Identities can be used to simplify trigonometric expressions.
Simplify.
c os sin tan
cos sin
sin
cos
cos
sin2
cos
cos 2 sin2
cos
1
cos
sec
a)
b)
cot2
1 sin2
cos 2
sin2
cos 2
1
1
sin2
csc2
5.4.5
cos2
sin2
1
cos2
14. 5.4.6
Simplifing Trigonometric Expressions
c) (1 + tan x)2 - 2 sin x sec x
1 2 tanx tan2 x 2
sinx
cosx
1 tan2 x 2tanx 2tanx
sec2 x
d)
cscx
tanx cotx
1
sinx
sinx
cosx
cosx
sinx
1
sinx
sin2 x cos 2 x
sinxcos x
1
sinx
1
sinx cosx
1
sinx
sinx cos x
1
cos x
(1 tanx)2 2 sinx
1
cosx
15. 5.4.7
Proving an Identity
Steps in Proving Identities:
1. Start with the more complex side of the identity and work
with it exclusively to transform the expression into the
simpler side of the identity.
2. Look for algebraic simplifications:
• Do any multiplying , factoring, or squaring which is
obvious in the expression.
• Reduce two terms to one, either add two terms or
factor so that you may reduce.
16. 16
3. Look for trigonometric simplifications
• Look for familiar trig relationships :
• If the expression contains squared terms
• , think of the Pythagorean Identities.
Transform each term to sine or cosine, if the
expression cannot be simplified easily using
other ratios.
17. 5.4.8
Proving an Identity
Prove the following:
a) sec x(1 + cos x) = 1 + sec x
= sec x + sec x cos x
= sec x + 1
1 + sec x
L.S. = R.S.
b) sec x = tan x csc x
sinx
cos x
1
sinx
1
cosx
secx
secx
L.S. = R.S.
c) tan x sin x + cos x = sec x
sinx
cosx
sinx
1
cosx
sin2 x cos 2 x
cos x
1
cosx
secx
secx
L.S. = R.S.
19. Proving an Identity
5.4.10
f)
cosA
1 sinA
1 sinA
cos A
2 secA
cos2 A (1 sinA)(1 sinA)
(1 sinA)(cosA)
cos2 A (1 2sinA sin2 A)
(1 sinA)(cosA)
cos2 A sin2 A1 2sinA
(1 sinA)(cosA)
2 2sinA
(1 sinA)(cosA)
2(1 sinA)
(1 sinA)(cosA)
2
(cosA)
2secA
2secA
L.S. = R.S.
20. Using Exact Values to Prove an Identity
5.4.11
Consider sinx
1 cos x
1 cosx
sinx
.
a) Use a graph to verify that the equation is an identity.
b) Verify that this statement is true for x =
6
.
c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions.
y
1 sinx
cosx
1 sinx
cosx
a)
21. Using Exact Values to Prove an Identity [cont’d]
b) Verify that this statement is true for x =
sinx
1 cosx
1 cosx
sinx
1
2
1
3
2
6
.
sin
6
1 cos
6
1
2
2
2 3
1
2 3
1 cos
6
sin
6
3
2
1
2
1
2 3
2
2
1
2 3
2 3
Rationalize the
denominator:
1
2 3
1
2 3
2 3
2 3
2 3
4 3
2 3
L.S. = R.S.
Therefore, the identity is
true for the particular
case of x
5.4.12
6
.
22. Using Exact Values to Prove an Identity [cont’d]
c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions.
5.4.13
sinx
1 cosx
1 cosx
sinx
sinx
1 cosx
1 cosx
1 cosx
sinx(1 cosx)
1 cos2 x
sinx(1 cosx)
sin2 x
1 cosx
sinx
1 cosx
sinx
L.S. = R.S.
Restrictions:
Note the left side of the
equation has the restriction
1 - cos x ≠ 0 or cos x ≠ 1.
Therefore, x ≠ 0 + 2 n,
where n is any integer.
The right side of the
equation has the restriction
sin x ≠ 0. x = 0 and
Therefore, x ≠ 0 + 2n
and x ≠ + 2n, where
n is any integer.
23. Proving an Equation is an Identity
Consider the equation sin2 A1
sin2 A sinA
1
1
sinA
.
a) Use a graph to verify that the equation is an identity.
b) Verify that this statement is true for x = 2.4 rad.
c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions.
y
sin2 A 1
sin2 A sinA
1
1
sinA
a)
5.4.14
24. Proving an Equation is an Identity [cont’d]
b) Verify that this statement is true for x = 2.4 rad.
sin2 A1
sin2 A sinA
1
1
sinA
(s in 2.4)2 1
(s in 2.4)2 sin2.4
= 2.480 466
1
1
sin 2.4
= 2.480 466
L.S. = R.S.
Therefore, the equation is true for x = 2.4 rad.
5.4.15
25. 5.4.16
Proving an Equation is an Identity [cont’d]
c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions.
sin2 A1
sin2 A sinA
1
1
sinA
(s inA1)(sinA 1)
sinA(s inA1)
(sinA1)
sinA
sinA
sinA
1
sinA
1
1
sinA
1
1
sinA
L.S. = R.S.
Note the left side of the
equation has the restriction:
sin2A - sin A ≠ 0
sin A(sin A - 1) ≠ 0
sin A ≠ 0 or sin A ≠ 1
A 0, or A
2
Therefore A, 0 2 n or
A + 2n, or
A
2
2 n, wheren is
any integer.
The right side of the
equation has the restriction
sin A ≠ 0, or A ≠ 0.
Therefore, A ≠ 0, + 2 n,
where n is any integer.
26. Applications of Trigonometry
This field of mathematics can be applied in
astronomy,navigation, music theory, acoustics, optics,
analysis of financial markets, electronics, probability
theory, statistics, biology, medical imaging (CAT scans
and ultrasound), pharmacy, chemistry, number theory
(and hence cryptology), seismology, meteorology,
oceanography and in many physical sciences.
26
27. Trigonometry is a branch of Mathematics
with several important and useful
applications. Hence it attracts more and
more research with several theories
published year after year
27
Conclusion