Question

Find the exact value of cos(15 degrees).

a) ((√3 - 1)/(2√2))
b) ((√2 + √6)/4)
c) ((√6 - √2)/4)
d) ((√3 + 1)/(2√2))

Answer 1

To find the exact value of cos(15 degrees), we use the difference formula for cosine, with known values for cos(45 degrees) and cos(30 degrees). The calculation shows that cos(15 degrees) equals ((√6 + √2)/4), corresponding to option b.

Step-by-step explanation:

To find the exact value of cos(15 degrees), we can use the half-angle identity for cosine, which is derived from the sum and difference formulas for cosine. The half-angle identity states that cos(θ/2) = ±√((1 + cos(θ))/2). However, for 15 degrees, we don't have a direct half-angle, but we do know that 15 degrees is 45 degrees minus 30 degrees, so we can use the difference formula for cosine:

• cos(45 degrees - 30 degrees) = cos(45 degrees)cos(30 degrees) + sin(45 degrees)sin(30 degrees)

Since cos(45 degrees) = sin(45 degrees) = √2/2 and cos(30 degrees) = √3/2 and sin(30 degrees) = 1/2, substituting these values we get:

• cos(15 degrees) = (√2/2)(√3/2) + (√2/2)(1/2)
• = (√6 + √2)/4

Therefore, the exact value of cos(15 degrees) is ((√6 + √2)/4), which corresponds to option b.