More Math Problems

Question

Use appropriate sum or difference identities to find the exact value.

cos 15°

Answer 1

You start with this trigonometric identity:

cosθcosΦ - sinθsinΦ = cos(θ+Φ)

Now if θ and Φ are both equal to 15⁰,

cos15⁰cos15⁰ - sin15⁰sin15⁰ = cos(15⁰+15⁰)

cos²15⁰ - sin²15⁰ = cos(30⁰)

now we use another trigonmetric identity to express the square of the sine of 15⁰

in terms of the cosine

cos²θ + sin²θ = 1 => sin²θ = 1 - cos²θ

so sin²15⁰ = 1 - cos²15⁰

and putting this in the other equation, we have

cos²15⁰ -(1 - cos²15⁰ ) = cos(30⁰)

2cos²15⁰ - 1 = cos(30⁰)

2cos²15⁰ = 1 + cos(30⁰)

cos²15⁰ = (1/2)(1+ cos(30⁰))

cos15⁰ = sqrt((1/2)+(sqrt(3)/2))