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Trigonometric Identities

Answer the following questions using what you’ve learned from this lesson. Write your responses in the space provided and turn the assignment in to your instructor.

 

1.     Derive this identity from the sum and difference formulas for cosine:

sin a sin b = (1 / 2)[cos(a – b) – cos(a + b)]

Start with the right-hand side since it is more complex.

 

2.     Use the trigonometric subtraction formula for sine to verify this identity:

sin((π / 2) – x) = cos x

 

 

 

3.     Find the exact value of each of the following:

a.               cos(7π / 12)

b.               tan(17π / 12)

c.               sin(195°)

d.               cot(15°)

Answer:

 

 

4.     Rewrite sin⁴ x so that it involves only the first power of cosine.

Answer:

 

5.     Prove the identity:

(sin(2x) / sin x) – (cos(2x) / cos x) = sec x

Answer:

 

6.     Prove the identity:

(cos x + cos y)² + (sin x – sin y)² = 2 + 2cos(x + y)

Answer:

 

7.    Solve these equations graphically on the interval [0, 2π]. Sketch the graph and list the solutions.

a. sin x – 1 = cos x

b. sin(2x) – 1 = tan x

Answer:

 

8.     Solve these equations:

a. sin(x + (π / 4)) – sin(x – (π / 4) = 1

b. sin x cos x = √3 / 4

c. tan² x – 3tan x + 2 = 0

 

Answer: