Practice with Trigonometric Identities - Rochester Institute of Technology

[Pages:2]Practice with Trigonometric Identities

Complete the following practice exercise using the Trigonometry Identities reference page handout.

Practice Exercises:

1. Given the double angle formula for cosine: cos 2 cos2 sin2

a. Use the trig identity sin2 cos2 1 to rewrite cos 2 in terms of sin2 only.

b. Use the trig identity sin2 cos2 1 to rewrite cos 2 in terms of cos2 only.

2. Consider the following identity: sin cos 2

a. Draw the graph of the cosine function on the domain 0 2 .

b. Extend the graph of the cosine function to show the graph for 0 . 2

c. How does the graph of the cosine function for 3 compare to the sine function

2

2

for 0 2 ?

d. Verify that the identity above: sin cos is correct using the addition identity for 2

sin(x y) .

3. Derive the double angle formula for sine by using the addition formula for sine. That is, find

sin2x sinx x etc. Derive the double angle formula for cosine using a similar technique.

4. Derive the half angle formula for sin2 x by starting with the cosine double angle formula cos 2x 1 2sin2 x and by solving for sin2 x in terms of cos 2x . Derive the other half angle formula using a similar technique.

5. The Law of Sines and Cosines are applicable to all triangles. Find the length of side "a" of triangle ABC if:

a. A 40 , B 100 , b 20

(Use Law of Sines)

b. A 40 , c 12 , b 20

(Use Law of Cosines)

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SOLUTIONS: 1. a) cos 2 cos2 sin2

1 sin2 sin2

1 2sin2

b) cos 2 cos2 sin2

cos2 1 cos2

2cos2 1

y cos

y cos

2. a)

b)

2

c) The graph is the same as one period of sine (this identity states that the cosine function is

the same as the sine function shifted units to the left). 2

d) sin(x y) sin xcos y cos xsin y

sin sin cos cos sin

2

2

2

sin 0 cos 1

cos

3. sin 2x sinx x sin xcos x cos xsin x 2sin xcos x cos 2x cosx x cos x cos x sin xsin x cos2 x sin2 x

4. cos 2x 1 2sin2 x 2sin2 x 1 cos 2x

sin 2 x 1 1 cos 2x

2

cos 2x 2cos2 x 1 2cos2 x 1 cos 2x

cos2 x 1 1 cos 2x

2

5. a) sin 40 sin100

a

20

a 13.05

b) a2 202 122 2012cos 40

a 18.98

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