Trigonometric identities

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1 Trigonometric Identities Peggy Adamson Mathematics Learning Centre University of Sydney NSW 006 c 986 University of Sydney

2 Contents Introduction. How to use this book..... Introduction Objectives Pretest.... Relations between the trigonometric functions 3 The Pythagorean identities 4 4 Sums and differences of angles 7 5 Double angle formulae 6 Applications of the sum, difference, and double angle formulae 7 Self assessment 3 8 Solutions to exercises 4

3 Mathematics Learning Centre, University of Sydney Introduction. How to use this book You will not gain much by just reading this booklet. Have pencil and paper ready to work through the examples before reading their solutions. Do all the exercises. It is important that you try hard to complete the exercises on your own, rather than refer to the solutions as soon as you are stuck.. Introduction This unit is designed to help you learn, or revise, trigonometric identities. You need to know these identities, and be able to use them confidently. They are used in many different branches of mathematics, including integration, complex numbers and mechanics. The best way to learn these identities is to have lots of practice in using them. So we remind you of what they are, then ask you to work through examples and exercises. We ve tried to select exercises that might be useful to you later, in your calculus unit of study..3 Objectives By the time you have worked through this workbook you should be familiar with the trigonometric functions sin, cos, tan, sec, csc and cot, and with the relationships between them, know the identities associated with sin θ + cos θ, know the expressions for sin, cos, tan of sums and differences of angles, be able to simplify expressions and verify identities involving the trigonometric functions, know how to differentiate all the trigonometric functions, know expressions for sin θ, cos θ, tan θ and use them in simplifying trigonometric functions, know how to reduce expressions involving powers and products of trigonometric functions to simple forms which can be integrated.

4 Mathematics Learning Centre, University of Sydney.4 Pretest We shall assume that you are familiar with radian measure for angles, and with the definitions and properties of the trigonometric functions sin, cos, tan. This test is included to help you check how well you remember these.. Express in radians angles of i. 60 ii. 35 iii. 70. Express in degrees angles of π i. ii. 3π 4 iii. π 3. What are the values of i. sin π ii. cos 3π iii. tan 3π 4 iv. sin 7π 6 V. cos 5π 3 vi. tan π 4. Sketch the graph of y cos x. Relations between the trigonometric functions Recall the definitions of the trigonometric functions by means of the unit circle, x + y. sin θ y (x, y) cos θ x θ tan θ y x Three more functions are defined in terms of these, secant (sec), cosecant (cosec or csc) and cotangent (cot). sec θ csc θ cot θ cos θ sin θ tan θ () () (3)

5 Mathematics Learning Centre, University of Sydney 3 The functions cos and sin are the basic ones. Each of the others can be expressed in terms of these. In particular tan θ sin θ cos θ cot θ cos θ sin θ (4) (5) These relationships are identities, not equations. An equation is a relation between functions that is true only for some particular values of the variable. For example, the relation sin θ cos θ is an equation, since it is satisfied when θ π 4, but not for other values of θ between 0 and π. On the other hand, tan θ sin θ cos θ is true for all values of θ, so this is an identity. The relationships () to (5) above are true for all values of θ, and so are identities. They can be used to simplify trigonometric expressions, and to prove other identities. Usually the best way to begin is to express everything in terms of sin and cos. Examples. Simplify the function cos x tan x. cos x tan x cos x sin x cos x sin x sin θ + tan θ. Show that sin θ tan θ. csc θ + cot θ To show that an identity is true, we have to prove that the left hand side and the right hand side are different ways of writing the same function. We usually do this by starting with one side and using the identities we know to transform it until we obtain the expression on the other side. sin θ + tan θ csc θ + cot θ sin θ + sin θ cos θ + cos θ sin θ sin θ (sin θ cos θ + sin θ) + cos θ sin θ cos θ sin θ( + cos θ) cos θ( + cos θ)

6 Mathematics Learning Centre, University of Sydney 4 sin θ cos θ sin θ tan θ Exercises. Simplify a. sin x cot x b. csc θ sec θ sin x + tan x c. + sec x. Show that cot θ + a. cot θ + tan θ tan θ b. cot x + sin x + cos x csc x c. ( + tan x) sin x sin x + cos x tan x. 3 The Pythagorean identities Remember that Pythagoras theorem states that in any right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. In the right angled triangle OAB, x cos θ and y sin θ, so cos θ + sin θ (6). A(x, y) y O θ x B

7 Mathematics Learning Centre, University of Sydney 5 Remember that cos θ means (cos θ) cos θ cos θ. Two other important identities can be derived from this one. Dividing both sides of (6) by cos θ we obtain cos θ cos θ + sin θ cos θ cos θ ie + tan θ sec θ. If we divide both sides of (6) by sin θ we get ie cot θ + csc θ. Summarising, cos θ sin θ + sin θ sin θ sin θ cos θ + sin θ (6) + tan θ sec θ (7) cot θ + csc θ (8) Examples. Simplify the expression sec θ sec θ. sec θ sec θ sec θ tan θ cos θ sin θ cos θ sin θ. Show that cos θ sin θ cos θ csc θ. tan θ cot θ. tan θ cot θ sin θ cos θ cos θ sin θ sin θ cos θ sin θ cos θ cos θ sin θ cos θ.

8 Mathematics Learning Centre, University of Sydney 6 Exercises. Simplify a. tan x + cot x b. ( sin t)( + tan t) c. + cos θ sec θ tan θ + cos θ sec θ + tan θ.. Show that a. sin 4 θ cos 4 θ cos θ b. tan x csc x tan x sin x + cos x c. + sec θ tan θ tan θ sec θ. Remember that you used these identities in finding the derivatives of tan, sec, csc and cot. Recall that d d (sin x) cos x and (cos x) sin x. dx dx Then d d (tan x) dx dx ( ) sin x cos x cos x cos x sin x( sin x) cos x cos x + sin x cos x cos x sec x. Exercises 3 Find. d dx (cot x),. d dx (sec x), 3. d (csc x). dx

9 Mathematics Learning Centre, University of Sydney 7 4 Sums and differences of angles A number of useful identities depend on the expressions for sin(α + β) and cos(α β). We shall state these expressions, then show how they can be derived. sin(α + β) sin α cos β + cos α sin β (9) cos(α + β) cos α cos β sin α sin β (0) sin(α β) sin α cos β cos α sin β () cos(α β) cos α cos β + sin α sin β () The expressions for sin(α + β), sin(α β) and cos(α + β) can all be derived from the expression for cos(α β). We derive that expression first. Look at the two diagrams below containing the angle (α β). We assume α is greater than β. We draw α and β in standard position (ie from the positive x-axis), and let A and B be the points where the terminal sides of α and β cut the unit circle. We draw the angle α β in standard position and let A be the point where its terminal side cuts the unit circle. A A' B α α β O β O α β B' A is the point (cos α, sin α). B is the point (cos β,sin β). A is the point (cos(α β), sin(α β)). B is the point (, 0). The triangles OAB and OA B are congruent, since triangle OA B is obtained by rotating OAB until OB lies along the x-axis. Therefore AB and A B are equal in length. Recall that the distance between two points P(x,y ) and Q(x,y ) is given by the formula So the distance AB is given by (PQ) (x x ) +(y y ). (AB) (cos β cos α) + (sin β sin α) cos β cos α cos β + cos α + sin β sin α sin β + sin α cos α cos β sin α sin β.

10 Mathematics Learning Centre, University of Sydney 8 The distance A B is given by (A B ) (cos(α β) ) + (sin(α β)) cos (α β) cos(α β)++sin (α β) These distances are equal so cos(α β). cos(α β) cos α cos β sin α sin β cos(α β) cos α cos β + sin α sin β. From this we can derive expressions for cos(α + β), sin(α + β) and sin(α β). In order to do this we need to know the following results: sin( θ) sin θ cos( θ) cos θ (x,y) O θ _ θ (x, y) and sin(θ) cos( π θ) cos(θ) sin( π θ). π _ θ θ Now cos(α + β) cos(α ( β)) cos α cos( β) + sin α sin( β) cos α cos β sin α sin β

11 Mathematics Learning Centre, University of Sydney 9 sin(α + β) cos[ π (α + β)] cos[( π α) β] cos( π α) cos β + sin(π α) sin β sin α cos β + cos α sin β sin(α β) sin(α +( β)) sin α cos( β) + cos α sin( β) sin α cos β cos α sin β. These formulae can be used in many different ways. Examples. Simplify sin(a + b) + sin(a b). sin(a + b) + sin(a b) sin a cos b + cos a sin b + sin a cos b cos a sin b sin a cos b.. Prove sin( π + θ) cos θ using the addition formulae. sin( π + θ) sin π cos θ + cos π sin θ cos θ +0 sin θ. cos θ. Exercises 4. Simplify a. b. c. sin(a + B) sin(a B) sin A sin B cos(a + B) + cos(a B) cos(a + B) cos(a B). cos A sin B

12 Mathematics Learning Centre, University of Sydney 0. Prove a. sin(π θ) sin θ b. cos(π θ) cos θ c. cos( π θ) sin θ d. cos( π + θ) sin θ. Expressions for tan(a + B) and tan(a B) follow in a straightforward way. Try to derive them for yourself first. tan(a + B) sin(a + B) cos(a + B) sin A cos B + cos A sin B sin A sin B sin A cos B + cos A sin B sin A sin B tan A + tan B tan A tan B. Summary tan(a B) sin(a B) cos(a B) sin A cos B cos A sin B + sin A sin B sin A cos B cos A sin B + sin A sin B tan A tan B + tan A tan B. tan(a + B) tan(a B) tan A + tan B tan A tan B tan A tan B + tan A tan B (3) (4) Exercises 5. Show that cot(α + β) cot α cot β cot α + cot β.. Setting α π 3 and β π 3, write down values of tan α, tan β and verify the expressions for tan(α + β) and tan(α β).

13 Mathematics Learning Centre, University of Sydney 5 Double angle formulae Expressions for the trigonometric functions of θ follow very easily from the preceding formulae. We shall summarise them and ask you to derive them as an exercise. sin θ sin θ cos θ (5) cos θ cos θ sin θ (6) cos θ cos θ (7) Example Show cos θ cos θ. cos θ sin θ (8) tan θ tan θ tan θ cos θ cos(θ + θ) cos θ cos θ sin θ sin θ cos θ sin θ cos θ ( cos θ) cos θ. Exercise Derive the rest of the expressions above. Example sin θ Simplify (9) cos θ. sin θ cos θ sin θ cos θ ( sin θ) sin θ cos θ sin θ cot θ. Exercises 6. Simplify + sin( π x) sin( π x).. Simplify 3. Simplify + cos θ. sin θ + sin A cos A cos A + sin A.

14 Mathematics Learning Centre, University of Sydney 6 Applications of the sum, difference, and double angle formulae A number of relations which are very useful in integration follow from the identities in sections 4 and 5. From (7) cos θ cos θ it follows that cos θ ( + cos θ) (0) and from (5) cos θ sin θ it follows that sin θ ( cos θ) () These identities are very useful in integration. For example cos θdθ ( + cos θ)dθ θ + sin θ +C 4 so you need to be expert in using them to simplify expressions. Example Show that sin x cos x ( cos 4x). 8 sin x cos x ( cos x) ( + cos x) 4 ( cos x) 4 ( ( + cos 4x)) 4 ( cos 4x) ( cos 4x). 8 Exercises 7 Simplify. cos 4 3θ. sin 4 θ.

15 Mathematics Learning Centre, University of Sydney 3 We showed earlier that sin(a + B) + sin(a B) sin A cos B, so sin A cos B (sin(a + B) + sin(a B)). Obtain similar expressions for sin A sin B, and by using the expressions for cos(a + B) and cos(a B). These relationships are also useful in integration. Summary sin A cos B (sin(a + B) + sin(a B)) () cos A sin B (sin(a + B) sin(a B)) (3) (cos(a + B) + cos(a B)) (4) sin A sin B (cos(a B) cos(a + B)) (5) Example Find sin 6x cos xdx. sin 6x cos x (sin 8x + sin 4x)dx 6 cos 8x cos 4x +C 8 Exercises 8 Express as sums or differences the following products:. sin 7x cos 3x. cos 8x cos x 3. cos 6x sin 5x 4. sin 4x sin x. 7 Self assessment. Simplify sin θ csc θ sin θ + cos θ.. Simplify sin θ + sin θ tan θ. tan θ 3. Simplify sin( 3π + θ). 4. Verify cos 4 θ sin 4 θ cos θ. 5. Verify sin(a + B) + sin(a B) sin(a + B) sin(a B) tan A cot B.

16 Mathematics Learning Centre, University of Sydney 4 8 Solutions to exercises Pretest. a. π 3 b. 3π 4 c. 3π. a. 45 b. 70 c a. b. 0 c. d. e. f A graph of the function y cos x Exercises. a. cos x b. cot θ c. sin x Exercises. a. sin x cos x b. c. + tan θ Exercises 3 d. dx cot x csc x. d sec x sec x tan x dx d 3. csc x csc x cot x dx Exercises 4. a. cot A b. c. tan A Exercises 5. tan α 3 and tan β 3

17 Mathematics Learning Centre, University of Sydney 5 Exercises 6. a. cot x b. cot θ c. tan A Exercises 7. (3 + 4 cos 6θ + cos θ) 8. (3 4 cos θ + cos 4θ) 8 Exercises 8. (sin 0x + sin 4x) (cos 0x + cos 6x) (sin x sin x) (cos x cos 6x) Self assessment sin θ csc θ. sin θ + cos θ. Use csc θ sin θ and sin θ + cos θ. sin θ + sin θ tan θ sec θ tan θ Use + tan θ sec θ, tan θ sin θ cos θ and sec θ cos θ. 3. sin( 3π + θ) cos θ Use sin 3π and cos 3π cos 4 θ sin 4 θ (cos θ sin θ)(cos θ + sin θ) cos θ cos θ. 5. sin(a + B) + sin(a B) sin(a + B) sin(a B) sin A cos B cos A sin B tan A cot B.

18 Mathematics Learning Centre T F E mlc.enquiries@sydney.edu.au sydney.edu.au/mlc Mathematics Learning Centre Produced by UPS, the University of Sydney, January 00. The University reserves the right to make alterations to any information contained within this publication without notice. ABN CRICOS 0006A

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Untitled-3 SEC.. Separable Equations In each of problems 1 through 8 solve the given differential equation : ü 1. y ' x y x y, y 0 fl y - x 0 fl y - x 0 fl y - x3 3 c, y 0 ü. y ' x ^ y 1 + x 3 x y 1 + x 3, y 0 fl