Chapter 5 Notes. 5.1 Using Fundamental Identities

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1 Chapter 5 Notes 5.1 Using Fundamental Identities 1. Simplify each expression to its lowest terms. Write the answer to part as the product of factors. (a) sin x csc x cot x ( 1+ sinσ + cosσ ) (c) 1 tanx secx. Evaluate: (a) x x + 81, where x = 9 tanθ ( 4 x ) 3, where x = sinθ 3. Verify: 1 cos x 1+ cos x = csc x cot x 4. If cos x+ sin x = cos x, show that cos x sin x= sin x. 5. Simplify: 1 1 sinθ sinθ. Verify: sin x + cos x sin x cos x + sin x cos x sin x + cos x = cos x sin x 7. Factor and simplify. sin x + cot xsin x + sin x cot 4 x 8. If the sum of the angles of a triangle x, y, and z is π. Simplify the expression sec x( sin( y + z) ). 9. Which of the following statements is/are true? Explain. (a) ( 1 cos x) =1 cos x ( 1 sin x) = 1 sin x 10. Find the value of cos 70 + cos110. Give an exact answer. Section 5. Verifying Trigonometric Identities 1. Prove: csc x + cot x = sin x 1 cos x

2 tan y. Verify: 1 cot y + cot y = 1+ sec ycsc y 1 tan y 3. Prove: ( 1+ sect) 3 ( 1+ cost) 3 = sec 3 t ( 1+ cost). 4. Prove: π sec x π tan x sec π x + tan π x = csc x( 1 cos x). 5. Prove: csc x cot x =1+ 3csc x cot x. Section 5.3 Solving Trigonometric Equations 1. Find all solutions of the given equation. (a) cos x 1 = 0 csc x = 0 (c) sec x 1sec x = cos x 1cos x (d) 1 = cos x + 5sin x (e) sint cos 3t = 0. Solve each equation on the interval [0, π ). Check for extraneous solutions. (a) sin 3x =1+ cos 3x (d) sec x + tan x = cos x 1 csc x 1+ sec x = 1+ csc x 1 sec x (c) tan x + tan x 3 = 0! 1 $! 1 $ 3. Solve the equation # 1+ &# 1 & = 0 on the interval [ π, π ]. " tan x %" cot x % 4. Solve the equation on the specified interval (a)sin x = 3, [ π, π ). 5 t 5. Solve the equation cot 3 = 0 on the interval ( 0, 8π ). cos( x) = 1 10, [ 0, π ]. Solve the equation cos 3 x 3cos x + 4 = 0.

3 5.4 Sum and Difference Formulas 1. Use the following graphs to derive the sum and difference formulas. Evaluate the following and give their exact answers when possible. (a) tan 11 1 π cos ( 15 ) 3. Write as a single trigonometric function and evaluate when possible (a) sin π cos π sin π cos π o o sin( 0 + y) cos y sin ycos ( 0 + y) 4. Find cos( u v), given that sinu = 1 3, 0 u π andsinv = 4 5, π < v 3π " 5. Simplify sin$ x + 7π # + y % '. &. Verify. cot x tan y = cos(x + y) sin xcos y 7. Solve the following equation on the interval 0 x 180 o 1 cos( x + 45 ) = 8. Express sin x + 8cos x as a function of sin x only. 9. Simplify each expression. (a) tan cos 1 ( 1) + arcsin x o o. ( ) sin( sin 1 θ + arctanθ ) 10. Find the exact value of arctan1+ arctan + arctan 3.

4 11. Express 1 sinx + 3 cosx in terms of sine only, and use your result to solve the equation 1 sinx + 3 cosx = Evaluate and simplify ( x + 4) 3 when x = tanθ. x 13. Solve the equation sin x + 3 cos x =1. Review: 14. What is the greatest value of cos 4x? 15. Prove: cos x 1 = sin x. 1+ sin x 1. Graph once cycle of y = sin x where π x π. Clearly label the relative extrema. 17. Find all the solutions of the equation 3sin x + cos x =. Hint: use problem #8 in this section. 18. Verify: cos x + π 4 + cos x π 4 = Find the exact value of arctan 1 + arctan 1 3. Section 7.3 Double and Half Angles, Product Sum Formulas 1. Use the following diagram to derive the identity sinθ = sinθ cosθ. The radius of the semicircle is 1 with center D.. Write as a single trigonometric function and evaluate when possible. tan 45 (a) sin4π cos4π sin 15 1 (c) 1 tan 45 π π (d) sin 3 8 cos3 8

5 3. Given that tanx = 1, show that tan x = ± Find sin θ, cos θ, tan θ if tanθ = 3 5, π < θ π 5. If cos x = p, q 0, find an expression for cosx in terms of p and q. q. If sinx = 1, find the value of cos 4x Simplify (a) sin arccos x 8. Prove: (a) 1 cos x = tan x sin x " ( ). sin$ arcsin π # % & 'cos " arcsin π % $ # ' & sin x + cos x sin x cos x + sin x cos x sin x + cos x = secx 9. Solve: 4 cos x 4 = csc x 4, 0 < x < 3π Half angle formulas 10. Find the exact value of: (a) sin.5 7π cos 8 (c) 1+ cos1θ (d) 1 cos x sin x x x x 11. Find the value of sin, cos, tan. (a) 3 o cos x=, 180 < x< 70 5 o 3π tan x= 1, < x< π 1. Use the triangle to find the value of cos θ.

6 13. Use the power reducing identities to write sin 4 x in terms of the first power of cosine only. Multiple angles are ok. 14. Evaluate 15. Evaluate 1 m 11π 5 cos cos where m = and n = 1 n π 5π x y where x= cos and y= sin 1 1 π. Product-to-sum and Sum-to-product Identities 1. Write sin8x+ sin x as a product. 17. Find the product: cos105 sin Verify sin 4x + cos 3x + sinx + cos x = cos x( sin 3x + cosx). 19. (a) Verify that sin x + 5π Solve the equation sin x + 5π sin x 5π sin x 5π = 1 4 ( 1 cos 4x ) = If cos x + cos 3x + cos5x cos(n 1)x = sinnx sin x sin x sin 3x + sin5x = sinx cos x, (a) Simplify: sin x sin 3x + sin5x cos x + cos 3x + cos5x. What is the value of cos15 + cos 45 + cos 75 + cos105 + cos135 + cos15?, where sin x 0 and n is a positive integer; and 1. (a) Verify sin x + sin y cos x + cos y = tan! x + y $ # & " % Find the value of sin π 1 + sin π 4 cos π 1 + cos π 4.. Simplify: sin 4 5π 8 cos4 5π Find the general solution of sec x = sec x cos x. 4. (a) Verify the identity cos 3x = cos x cosx cos x, by using the left hand side. Find the value of cos π 1 cos π cos π Simplify each expression. Give answers in simplest form.! (a) sin arccos 1 $! 1 # &, x < 0 sin " x % arctan 3 $ # & (c) " 4 % tan π 8 1+ tan π 8