The document discusses trigonometry concepts related to 3D shapes and solving problems involving angles of elevation. Specifically:
- When doing 3D trigonometry, it is often useful to redraw shapes in 2D to analyze them.
- A worked example problem is shown where a life raft and cabin cruiser positions are used to calculate the distance and bearing between them given elevation angle measurements of a mountain peak from both locations.
- Trigonometric relationships involving tangent, sine and cosine are used along with angle properties to set up and solve equations to find the distance as 2799m and bearing as 249°51'.
The document discusses capacitors and dielectrics. It contains examples and questions about how capacitance, charge, potential difference, and electric field are affected by changing the plate separation of a capacitor or inserting a dielectric. Key effects discussed include:
1) Increasing the plate separation of a capacitor decreases the capacitance and increases the potential difference required to maintain the same charge.
2) Inserting a dielectric between capacitor plates increases the capacitance and decreases the electric field strength inside. The potential difference and charge on the plates may increase, decrease, or stay the same depending on other factors.
3) The electric field is strongest where the plates are closest together and weakest where they are farther apart
Projection of lines(new)(thedirectdata.com)Agnivesh Ogale
The line AB is inclined at 40 degrees to the VP. End A is 20mm above the HP and 50mm from the VP. The front view measures 65mm and the vertical trace is 10mm above the HP. To determine:
1) The true length of AB is 65/cos40° = 75mm
2) The inclination to the HP (θ) is 30 degrees
3) The horizontal trace is 10mm above the VP
Projection of lines(new)(thedirectdata.com)Agnivesh Ogale
A 100mm line PQ is inclined at 30 degrees to the HP and 45 degrees to the VP. Its midpoint M is 20mm above the HP and in the VP. P is in the third quadrant and Q is in the first quadrant. The projections show PQ inclined at 30 degrees to the HP in front view and 45 degrees to the VP in top view, with midpoint M at 20mm above the HP in both views and the ends P and Q in their respective quadrants as given.
This document discusses the principles of engineering graphics related to projecting straight lines in orthographic projections. It begins by defining the different types of lines such as vertical, horizontal, and inclined lines. It then provides examples of how to determine the front view, top view, and true length of a line given information about its inclination to the planes and the positions of its ends. The document emphasizes that the front and top views of a line inclined to both planes will have reduced lengths. It concludes by discussing the special case of a line lying in a profile plane, where the front and top views will overlap on the same projector line.
This document lists three types of properties with their square footages: Type A is 1233 square feet, Type B is 1218 square feet, and Type C is 910 square feet.
The document provides information and instructions for drawing orthographic projections of points, lines, planes, and solids. It discusses the key elements needed, including a description of the object, observer location, and object placement relative to the horizontal and vertical planes. Guidelines are given for naming different views in projections using notations like a, a', and a" for the top, front, and side views of a point A. Examples are presented of drawing the projections of a point placed in different quadrants and of lines with different orientations. Methods are described for determining the true length and true inclinations to the planes when given the projections or when given other properties like the true length, inclinations, and object placement.
The document discusses capacitors and dielectrics. It contains examples and questions about how capacitance, charge, potential difference, and electric field are affected by changing the plate separation of a capacitor or inserting a dielectric. Key effects discussed include:
1) Increasing the plate separation of a capacitor decreases the capacitance and increases the potential difference required to maintain the same charge.
2) Inserting a dielectric between capacitor plates increases the capacitance and decreases the electric field strength inside. The potential difference and charge on the plates may increase, decrease, or stay the same depending on other factors.
3) The electric field is strongest where the plates are closest together and weakest where they are farther apart
Projection of lines(new)(thedirectdata.com)Agnivesh Ogale
The line AB is inclined at 40 degrees to the VP. End A is 20mm above the HP and 50mm from the VP. The front view measures 65mm and the vertical trace is 10mm above the HP. To determine:
1) The true length of AB is 65/cos40° = 75mm
2) The inclination to the HP (θ) is 30 degrees
3) The horizontal trace is 10mm above the VP
Projection of lines(new)(thedirectdata.com)Agnivesh Ogale
A 100mm line PQ is inclined at 30 degrees to the HP and 45 degrees to the VP. Its midpoint M is 20mm above the HP and in the VP. P is in the third quadrant and Q is in the first quadrant. The projections show PQ inclined at 30 degrees to the HP in front view and 45 degrees to the VP in top view, with midpoint M at 20mm above the HP in both views and the ends P and Q in their respective quadrants as given.
This document discusses the principles of engineering graphics related to projecting straight lines in orthographic projections. It begins by defining the different types of lines such as vertical, horizontal, and inclined lines. It then provides examples of how to determine the front view, top view, and true length of a line given information about its inclination to the planes and the positions of its ends. The document emphasizes that the front and top views of a line inclined to both planes will have reduced lengths. It concludes by discussing the special case of a line lying in a profile plane, where the front and top views will overlap on the same projector line.
This document lists three types of properties with their square footages: Type A is 1233 square feet, Type B is 1218 square feet, and Type C is 910 square feet.
The document provides information and instructions for drawing orthographic projections of points, lines, planes, and solids. It discusses the key elements needed, including a description of the object, observer location, and object placement relative to the horizontal and vertical planes. Guidelines are given for naming different views in projections using notations like a, a', and a" for the top, front, and side views of a point A. Examples are presented of drawing the projections of a point placed in different quadrants and of lines with different orientations. Methods are described for determining the true length and true inclinations to the planes when given the projections or when given other properties like the true length, inclinations, and object placement.
12X1 T04 05 approximations to roots (2011)Nigel Simmons
The document describes the method of halving intervals to find approximations of roots. It begins by stating if a function f(x) is continuous over an interval [a,b] where f(a) and f(b) have opposite signs, then there exists a root between a and b. It then works through an example of finding the root of x4 + 2x - 19 = 0 between 1 and 3 by repeatedly halving intervals and evaluating the function at the midpoint until reaching an approximation of 1.96 to two decimal places.
The document discusses conic sections, specifically the ellipse. An ellipse is defined as the locus of points where the sum of the distances from two fixed points (foci) is a constant. For an ellipse, the eccentricity e is less than 1. The foci of an ellipse lie on the major axis, and the distance from each focus to any point on the ellipse follows the relationship SA = eAZ. This allows deriving the equation of an ellipse as x2/a2 + y2/b2 = 1, where b2 = a2(1 - e2).
The document discusses graphing sine curve functions. It explains that a basic sine curve has the equation y = sin(x) with a period of 2π units. More generally, a sine curve has the equation y = a sin(bx + c) where the period is 2π/b units, the amplitude is a units, and c shifts the curve left or right. It provides an example of y = 5 sin(9x - π/2) with a period of 2π/9 units, amplitude of 5 units, and a right shift of π/18.
The document defines terminology and notation used in Euclidean geometry. It provides definitions for terms like parallel, perpendicular, congruent, and similar. It also describes conventions for naming angles, polygons, and parallel lines. The document concludes by outlining rules for constructing proofs in geometry, including stating given information, constructions, theorems, and logical reasoning.
The document discusses odd and even functions. Odd functions satisfy f(-x) = -f(x) and have point symmetry about the origin. Examples given include y = x3 and y = x7 - x5. Even functions satisfy f(-x) = f(x) and have line symmetry about the y-axis. Examples provided are y = x2 and y = x2 + 4. The key characteristics of odd and even functions are outlined and examples are used to prove that certain functions, such as y = x3 + x7, are odd functions.
The document describes two methods for finding the volume of a rectangular pyramid with a given height h, base length a, and base width b. The first method uses similar triangles, determining x and y in terms of z. The second method uses coordinate geometry and the slope formula to derive x = (b(h-z))/2h. The area of each incremental slice A(z) is calculated as ab(h-z)/h^2. Integrating this from 0 to h gives the total volume as ab(h-z)^2/2.
The document discusses the relationships between the roots and coefficients of polynomial equations. It provides formulas to find the sum of roots taken one, two, three, or more at a time in terms of the coefficients. For a polynomial of degree n in the form ax^n + bx^(n-1) + cx^(n-2) + ..., the formulas are given as:
- Sum of roots one at a time = -b/a
- Sum of roots two at a time = c/a
- Sum of roots three at a time = -d/a
- And so on for higher degrees.
The document discusses limits and continuity. It defines a limit as describing the behavior of a function as the input value approaches a particular number. It provides examples of calculating limits using direct substitution, factorizing, and special limits involving infinity. The key points covered are:
- A limit describes what value a function approaches as its input gets closer to a number
- Methods for calculating limits include direct substitution, factorizing, and using special rules for infinity
- A function is continuous if the left-hand and right-hand limits are equal at a point
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
12X1 T04 05 approximations to roots (2011)Nigel Simmons
The document describes the method of halving intervals to find approximations of roots. It begins by stating if a function f(x) is continuous over an interval [a,b] where f(a) and f(b) have opposite signs, then there exists a root between a and b. It then works through an example of finding the root of x4 + 2x - 19 = 0 between 1 and 3 by repeatedly halving intervals and evaluating the function at the midpoint until reaching an approximation of 1.96 to two decimal places.
The document discusses conic sections, specifically the ellipse. An ellipse is defined as the locus of points where the sum of the distances from two fixed points (foci) is a constant. For an ellipse, the eccentricity e is less than 1. The foci of an ellipse lie on the major axis, and the distance from each focus to any point on the ellipse follows the relationship SA = eAZ. This allows deriving the equation of an ellipse as x2/a2 + y2/b2 = 1, where b2 = a2(1 - e2).
The document discusses graphing sine curve functions. It explains that a basic sine curve has the equation y = sin(x) with a period of 2π units. More generally, a sine curve has the equation y = a sin(bx + c) where the period is 2π/b units, the amplitude is a units, and c shifts the curve left or right. It provides an example of y = 5 sin(9x - π/2) with a period of 2π/9 units, amplitude of 5 units, and a right shift of π/18.
The document defines terminology and notation used in Euclidean geometry. It provides definitions for terms like parallel, perpendicular, congruent, and similar. It also describes conventions for naming angles, polygons, and parallel lines. The document concludes by outlining rules for constructing proofs in geometry, including stating given information, constructions, theorems, and logical reasoning.
The document discusses odd and even functions. Odd functions satisfy f(-x) = -f(x) and have point symmetry about the origin. Examples given include y = x3 and y = x7 - x5. Even functions satisfy f(-x) = f(x) and have line symmetry about the y-axis. Examples provided are y = x2 and y = x2 + 4. The key characteristics of odd and even functions are outlined and examples are used to prove that certain functions, such as y = x3 + x7, are odd functions.
The document describes two methods for finding the volume of a rectangular pyramid with a given height h, base length a, and base width b. The first method uses similar triangles, determining x and y in terms of z. The second method uses coordinate geometry and the slope formula to derive x = (b(h-z))/2h. The area of each incremental slice A(z) is calculated as ab(h-z)/h^2. Integrating this from 0 to h gives the total volume as ab(h-z)^2/2.
The document discusses the relationships between the roots and coefficients of polynomial equations. It provides formulas to find the sum of roots taken one, two, three, or more at a time in terms of the coefficients. For a polynomial of degree n in the form ax^n + bx^(n-1) + cx^(n-2) + ..., the formulas are given as:
- Sum of roots one at a time = -b/a
- Sum of roots two at a time = c/a
- Sum of roots three at a time = -d/a
- And so on for higher degrees.
The document discusses limits and continuity. It defines a limit as describing the behavior of a function as the input value approaches a particular number. It provides examples of calculating limits using direct substitution, factorizing, and special limits involving infinity. The key points covered are:
- A limit describes what value a function approaches as its input gets closer to a number
- Methods for calculating limits include direct substitution, factorizing, and using special rules for infinity
- A function is continuous if the left-hand and right-hand limits are equal at a point
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
Thinking of getting a dog? Be aware that breeds like Pit Bulls, Rottweilers, and German Shepherds can be loyal and dangerous. Proper training and socialization are crucial to preventing aggressive behaviors. Ensure safety by understanding their needs and always supervising interactions. Stay safe, and enjoy your furry friends!
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
Physiology and chemistry of skin and pigmentation, hairs, scalp, lips and nail, Cleansing cream, Lotions, Face powders, Face packs, Lipsticks, Bath products, soaps and baby product,
Preparation and standardization of the following : Tonic, Bleaches, Dentifrices and Mouth washes & Tooth Pastes, Cosmetics for Nails.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
2. 3D Trigonometry
When doing 3D trigonometry it is often useful to redraw all of the
faces of the shape in 2D.
3. 3D Trigonometry
When doing 3D trigonometry it is often useful to redraw all of the
faces of the shape in 2D.
2003 Extension 1 HSC Q7a)
David is in a life raft and Anna is in a cabin cruiser searching for him.
They are in contact by mobile phone. David tells Ana that he can see
Mt Hope. From David’s position the mountain has a bearing of 109 ,
and the angle of elevation to the top of the mountain is 16.
Anna can also see Mt Hope. From her position it has a bearing of 139,
and and the top of the mountain has an angle of elevation of 23 .
The top of Mt Hope is 1500 m above sea level.
Find the distance and bearing of the life raft from Anna’s position.
19. H H
1500 m N’ 1500 m
16 23
D B A B
A 139
N
b
1500 tan 67 N”
D
109 30
1500 tan 74
B
20. b 2 1500 2 tan 2 67 1500 2 tan 2 74 2 1500 tan 67 1500 tan 74 cos 30
21. b 2 1500 2 tan 2 67 1500 2 tan 2 74 2 1500 tan 67 1500 tan 74 cos 30
b 2798.96
2799 (to nearest metre)
Anna and David are 2799 m apart.
22. H H
1500 m N’ 1500 m
16 23
D B A B
A 139
N
b
1500 tan 67 N”
D
109 30
1500 tan 74
B
23. b 2 1500 2 tan 2 67 1500 2 tan 2 74 2 1500 tan 67 1500 tan 74 cos 30
b 2798.96
2799 (to nearest metre)
Anna and David are 2799 m apart.
sin DAB sin 30
1500 tan 74 b
24. b 2 1500 2 tan 2 67 1500 2 tan 2 74 2 1500 tan 67 1500 tan 74 cos 30
b 2798.96
2799 (to nearest metre)
Anna and David are 2799 m apart.
sin DAB sin 30
1500 tan 74 b
1500 tan 74 sin 30
sin DAB
b
DAB 699' or 11051'
25. b 2 1500 2 tan 2 67 1500 2 tan 2 74 2 1500 tan 67 1500 tan 74 cos 30
b 2798.96
2799 (to nearest metre)
Anna and David are 2799 m apart.
sin DAB sin 30
1500 tan 74 b
1500 tan 74 sin 30
sin DAB
b
DAB 699' or 11051'
If DAB 699'
then BDA 8051'
26. b 2 1500 2 tan 2 67 1500 2 tan 2 74 2 1500 tan 67 1500 tan 74 cos 30
b 2798.96
2799 (to nearest metre)
Anna and David are 2799 m apart.
sin DAB sin 30
1500 tan 74 b
1500 tan 74 sin 30
sin DAB
b
DAB 699' or 11051'
If DAB 699'
then BDA 8051'
But DAB BDA
BDA 11051'
27. H H
1500 m N’ 1500 m
16 23
D B A B
A 139
N
b 11051'
1500 tan 67 N”
D
109 30
1500 tan 74
B
28. b 2 1500 2 tan 2 67 1500 2 tan 2 74 2 1500 tan 67 1500 tan 74 cos 30
b 2798.96
2799 (to nearest metre)
Anna and David are 2799 m apart.
sin DAB sin 30
1500 tan 74 b
1500 tan 74 sin 30
sin DAB
b
DAB 699' or 11051'
If DAB 699'
then BDA 8051'
But DAB BDA
BDA 110 51'
The bearing of David from Anna is 24951'
29. 2000 Extension 1 HSC Q3c)
A surveyor stands at point A, which is due south of a tower OT of
height h m. The angle of elevation of the top of the tower from A is 45
The surveyor then walks 100 m due east to point B, from where she
measures the angle of elevation of the top of the tower to be 30
31. (i) Express the length of OB in terms of h.
T
h
30
0 B
OB
tan 60
h
OB h tan 60
32. (i) Express the length of OB in terms of h.
T
h
30
0 B
OB
tan 60
h
OB h tan 60
(ii) Show that h 50 2
33. (i) Express the length of OB in terms of h.
T
h
30
0 B
OB
tan 60
h
OB h tan 60
(ii) Show that h 50 2
T
h
A 45 0
34. (i) Express the length of OB in terms of h.
T
h
30
0 B
OB
tan 60
h
OB h tan 60
(ii) Show that h 50 2
T
h
A 45 0
ATO is isosceles
AO h
35. (i) Express the length of OB in terms of h.
T
h
30
0 B
OB
tan 60
h
OB h tan 60
(ii) Show that h 50 2
T O
h h tan 60
h
A 45 0 A B
100
ATO is isosceles
AO h
36. (i) Express the length of OB in terms of h.
T
h
30
0 B
OB
tan 60
h
OB h tan 60
(ii) Show that h 50 2
T O h 2 100 2 h 2 tan 2 60
h h tan 60
h
A 45 0 A B
100
ATO is isosceles
AO h
37. (i) Express the length of OB in terms of h.
T
h
30
0 B
OB
tan 60
h
OB h tan 60
(ii) Show that h 50 2
T O h 2 100 2 h 2 tan 2 60
h h tan 60 h 2 100 2 3h 2
h
45 2h 2 100 2
A 0 A B 100 2
100 h2
ATO is isosceles 2
100
AO h h
2
h 50 2
39. (iii) Calculate the bearing of B from the base of the tower.
N
O
50 2
A 100 B
40. (iii) Calculate the bearing of B from the base of the tower.
N
100
tan AOB
50 2
O
50 2
A 100 B
41. (iii) Calculate the bearing of B from the base of the tower.
N
100
tan AOB
50 2
AOB 54 44'
O
50 2
A 100 B
42. (iii) Calculate the bearing of B from the base of the tower.
N
100
tan AOB
50 2
AOB 54 44'
O
bearing 180 54 44'
50 2 12516'
A 100 B
43. (iii) Calculate the bearing of B from the base of the tower.
N
100
tan AOB
50 2
AOB 54 44'
O
bearing 180 54 44'
50 2 12516'
A 100 B
Book 2: Exercise 2G odds
Exercise 2H evens