Lesson 7.3 Exercises, pages

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1 Lesson 7. Exercises, pages 8 A. Write each expression in terms of a single trigonometric function. cos u a) b) sin u cos u cot U tan U P DO NOT COPY. 7. Reciprocal and Quotient Identities Solutions 7

2 c) sin u sec u cos u csc u d) sin u tan u (sin U) a () b a b sin U sin U cos U cos U 4. Determine the non-permissible values of u. a) sec u b) tan u sec U, so, so non-permissible non-permissible values values occur when 0, occur when 0, so U k, k ç so U k, k ç csc u c) d) sec u cos u csc U, so sec U, so non-permissible non-permissible values occur values occur when 0, so when 0, so U k, U k, k ç ; or when k ç ; or when 0, 0, so U k, k ç so U k, k ç 5. Verify each identity for the given value of u. a) tan u csc u sec u sec u; u 50 Substitute: U 50 L.S. csc U sec U (tan 50 )(csc 50 )(sec 50 ) (tan 0 )a sin 50 b a cos 50 b (tan 0 ) a a sin 0 b cos 0 b a b() a b R.S. sec U sec (50 ) cos (50 ) a cos 0 b a b 4 4 right side, so U 50 is verified Reciprocal and Quotient Identities Solutions DO NOT COPY. P

3 tan u csc b) u cot u; u 4 sec u Substitute: U 4 L.S. csc U sec U atan 4 b a csc a 4 bb sec a 4 b atan bacos b asin b R.S. cot U 4 cot tan ( )a b a b right side, so U 4 is verified. B. Prove each identity in question 5. a) csc U sec U sec U L.S. csc U sec U a ba ba b cos U sec U R.S. right side, so the identity is csc b) U cot U sec U csc L.S. U sec U a ba b a b ()() a b cot U R.S. right side, so the identity is P DO NOT COPY. 7. Reciprocal and Quotient Identities Solutions 9

4 7. For each identity: i) Verify the identity using graphing technology. ii) Prove the identity. a) ()(csc u ) b) cot u - cot u - tan u i) Graph: y and y a b i) Graph: y and y The graphs coincide, so the identity The graphs coincide, so the is verified. identity is verified. cot U ii) R.S. ii) R.S. ()(csc U ) () a b L.S. ()( ) cot U L.S. right side, so the identity is 8. For each identity: i) Verify the identity for u 45. ii) Prove the identity. cot u a) csc u 0 b) tan u cos u sin u cos u csc u i) Substitute: U 45 cot U L.S. csc U cot 45 csc 45 cos 45 0 R.S. right side, so U 45 is verified. i) Substitute: U 45 L.S. tan U cos U sin U (tan 45 ) (cos 45 ) (sin 45 ) () a a b b R.S. csc U (csc 45 ) ( ) right side, so U 45 is verified Reciprocal and Quotient Identities Solutions DO NOT COPY. P

5 cot U ii) L.S. csc U 0 R.S. ii) L.S. tan U cos U sin U a sin U (cos U) sin U cos U b sin U sin U sin U csc U R.S. 9. For each identity: 7 i) Verify the identity for u. ii) Prove the identity. csc u - cos u - cot u a) csc u b) cot u i) Substitute: U L.S. csc U 7 csc csc csc U R.S. csc 7 7 i) Substitute: U cot U L.S. cos 7 cot 7 sin 7 a b sin 7 a b 7 right side, so U is verified. csc U ii) R.S. ()( ) csc U L.S. R.S. cot U 7 cot 7 right side, so U is verified. cot U ii) L.S. ()() ( )() ()( ) ( )() cot U R.S. P DO NOT COPY. 7. Reciprocal and Quotient Identities Solutions

6 0. Use algebra to solve each equation over the domain 0 x <. Give the roots to the nearest hundredth where necessary. a) tan x cot x b) cos x sin x 0 Assume tan x 0, then divide by Assume cos x 0, then divide tan x. by cos x. tan x cot x cos x sin x tan x tan x cos x cos x 0 tan x 0 tan x tan tan x x tan x 5 x or x 5 x, x, x, or For cos x 0, x or x 7 x 4 Verify by substitution that neither For tan x 0, x 0 or x value of x is a root of the given Verify by substitution that neither equation. value of x is a root of the given Verify the roots by substitution. equation. Verify the roots by substitution. c) cos x 7 sec x d) sin x sin x cos x cos x 7 cos x Multiply by cos x, then collect terms on one side. cos x 7 cos x 0 ( cos x )(cos x ) 0 Either cos x 0 cos x 5 x or x Or cos x 0 cos x This equation has no solution. Verify the roots by substitution. sin x sin x cos x 0 (sin x)(sin x cos x) 0 Either sin x 0 x 0 or x Or sin x cos x 0 sin x cos x Assume cos x 0, then divide by cos x. tan x 5 x or x 4 4 For cos x 0, x or x Verify by substitution that neither value of x is a root of the given equation. Verify the roots by substitution.. Identify any errors in this proof, then write a correct algebraic proof. To prove: - = Correct proof: csc u - L.S. R.S. - csc U - csc u - csc u - R.S. L.S. From the st line of the proof to the nd line, cannot be written as. From the 4th line of the proof to the 5th line, cannot be written as. csc U csc U 7. Reciprocal and Quotient Identities Solutions DO NOT COPY. P

7 . Identify which equation below is an identity. Justify your answer. Prove the identity. Solve the other equation over the domain x. Give the roots to the nearest hundredth. sin a) x + csc sin x b) x + + csc x sin x sin x csc x I graphed each side of the equation. The graphs do not coincide, so this is an equation. From the graphs, the roots are approximately: x 0.8 and x. I graphed each side of the equation. The graphs appear to coincide, so this is probably the identity. R.S. csc x csc x sin x sin x Multiply numerator and denominator by sin x. sin R.S. x sin x L.S. right side, so the identity is C. Here are two identities that involve the cotangent ratio: cos u cot u and cot u tan u a) Show how you can derive one identity from the other. cot U Write. cot U Multiply numerator and denominator by. cot U P DO NOT COPY. 7. Reciprocal and Quotient Identities Solutions

8 b) Determine the non-permissible values of u for each identity. Explain why these values are different. How could you illustrate this using graphing technology? For cot U For cot U Since, then 0 So, U k, k ç 0 and 0 So, U k, k ç and U k, k ç The values are different because there are two restrictions for and only one restriction for. When I graph y, and set the TABLE for intervals of, it shows ERROR for all values of X in the table. When I graph y, with the same TABLE settings, it shows ERROR only for X k, k ç Reciprocal and Quotient Identities Solutions DO NOT COPY. P