( ) + ( ) ( ) ( ) Exercise Set 6.1: Sum and Difference Formulas. β =, π π. π π. β =, evaluate tan β. Simplify each of the following expressions.
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1 Simplify each of the following expressions ( x cosx + cosx ( + x ( 60 θ + ( 60 + θ 6 cos( 60 θ + cos( 60 + θ 7 cosx + cosx+ 8 x+ + x ( θ 80 + ( θ cos( 90 + θ + cos( 90 θ 7 Given that tan ( α = and tan ( evaluate tan ( α + β 8 Given that tan ( α = and tan ( evaluate tan ( α β β =, Simplify each of the following expressions as much as possible without a calculator 9 cos( cos( 0 + ( ( 0 0 cos( 7 cos( ( 7 ( ( cos( cos( ( ( cos( 7 + cos( ( 7 cos cos cos cos 7 7 Given that tan ( α =, evaluate tan α + 6 cos cos cos cos + Given that tan ( Given that tan ( Given that tan ( evaluate tan β x =, evaluate tan x 7 x =, evaluate tan x + Given that tan ( x = and ( tan ( x + y tan y = 6, evaluate 7 ( A cos( A cos( A ( A 8 cos( α cos( α + ( α cos( α 9 0 ( + ( tan tan tan tan ( ( tan 9 tan + tan 9 tan 6 Given that tan ( x = and ( evaluate tan ( x y tan y =, Math 0, Precalculus
2 tan tan + tan tan 7 tan + tan 7 tan tan 8 For fractions with larger magnitude than those in Exercises 6 and 7, it can be helpful to use larger multiples of Rewrite each special angle below so that it has a denominator of (a (c 9 7 (b (d ( a b + tan ( b ( ab ( b tan tan tan ( c+ d ( d c ( c d ( d c tan tan + tan + tan Rewrite each special angle below so that it has a denominator of (a (d 6 (b (e (c (f 6 6 Use the answers from Exercise to write each fraction below as the sum of two special angles (Hint: Each of the solutions contains a multiple of (a (c 7 (b (d 7 Use the answers from Exercise to write each fraction below as the difference of two special angles (Hint: Each of the solutions contains a multiple of (a (c (b 7 (d 7 9 Use the answers from Exercises and 8 to write each fraction below as the sum of two special angles (Hint: Each of the solutions contains a multiple of (a (c 9 (b (d 0 Use the answers from Exercises and 8 to write each fraction below as the difference of two special angles (Hint: Each of the solutions contains a multiple of (a (c 9 (b (d 9 Use the answers from Exercises and 8, along with their negatives to write each fraction below as the difference, x y, of two special angles, where x is negative and y is positive 7 (a (b 9 (c (d Use the answers from Exercises and 8, along with their negatives to write each fraction below as the difference, x y, of two special angles, where x is negative and y is positive 9 (a (b Math 0, Precalculus
3 (c (d Use a sum or difference formula to prove that θ = θ Use a sum or difference formula to prove that cos θ = cos θ Use a sum or difference formula to prove that tan θ = tan θ 6 Use a sum or difference formula to prove that cos θ = ( θ Find the exact value of each of the following 7 cos( 8 ( 7 9 ( 9 0 cos( tan ( 0 tan ( cos 7 cos cos 9 tan Math 0, Precalculus tan 7 tan 9 tan (Hint: It may help to first draw right triangles in the appropriate quadrants and label the side lengths 6 Suppose that ( α = and ( β =, where 0 < α < < β < Find: (a ( α β (b cos( α + β (c tan ( α β 8 7 where 0 < β < α < Find: α + β 6 Suppose that cos( α = and cos( (a ( (b cos( α β (c tan ( α + β where < α < β < Find: α + β 6 Suppose that tan ( α = and cos( (a ( (b cos( α + β (c tan ( α β 66 Suppose that tan ( α = and ( where < α < and < β < Find: α β (a ( (b cos( α β (c tan ( α + β
4 Evaluate the following 67 cos tan 68 ( tan tan + tan tan cos tan tan Simplify the following 7 + cos 7 cos( A ( B cot ( B + tan ( A 7 tan ( A cos( B cos( A cos( A cot ( A tan ( B 7 ( A B ( A+ B 7 cos( A + B cos( A B Prove the following 7 ( x y + ( x+ y = ( x cos( y 76 cos( x y + cos( x+ y = cos( x cos( y cos ( x y + cos( x+ y ( x ( y ( x y + ( x+ y cos( x cos( y ( ( = cot = tan ( ( x y tan x tan y = x + y tan x + tan y ( ( cos x y + tan x tan y = cos x + y tan x tan y ( x cot( y ( x Use right triangle ABC below to answer the following questions regarding cofunctions, in terms of side lengths a, b, and c b A C a 8 (a Find ( A (b Find cos( B (c Analyze the answers for (a and (b What (d What is the relationship between angles A and B? (ie If you knew the measure of one angle, how would you find the other? Write (e Complete the following cofunction relationships by filling in blank Write your answers in terms of A cos ( A = cos ( ( A = ( 8 (a Find tan ( A (b Find cot ( B (c Analyze the answers for (a and (b What (d What is the relationship between angles A and B? (ie If you knew the measure of one angle, how would you find the other? Write (e Complete the following cofunction relationships by filling in blank Write your answers in terms of A tan ( A = cot ( cot ( A = tan ( 8 (a Find sec( A (b Find csc( B c B (c Analyze the answers for (a and (b What Continued on the next page Math 0, Precalculus
5 (d What is the relationship between angles A and B? (ie If you knew the measure of one angle, how would you find the other? Write (e Complete the following cofunction relationships by filling in the blank Write your answers in terms of A sec csc ( A = csc ( ( A = sec ( 97 sec( 0 ( cos csc 6 99 cot sec 00 ( cos 7 + cos ( 8 8 Use a sum or difference formula to prove that ( θ = cos( θ 90 Use cofunction relationships to solve the following for acute angle x 8 ( 7 = cos( x = 0 86 cos ( x = 87 sec csc( x 88 csc( = sec( x 89 tan ( 7 = cot ( x = 90 cot tan ( x Simplify the following 9 cos( 90 x csc( x 9 x sec( x 9 x tan ( x 9 csc x ( x 9 sec( 90 θ cos( θ 96 tan ( 90 x csc( 90 x Math 0, Precalculus
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