Use a sum or difference formula to find the exact value of cos5pi/12.

Answer 1

You can use either sum or difference formula

To make things simple we can convert 5π/12 to 75°

π=180° so (5 x 180)/12 = 75°

You can express cos75 as a sum of two angles, cos(45+30) or as a difference of two angles, cos(120-45)

Sum

cos(45+30)

√2/2 ∗√3/2 - √2/2 ∗ 1/2

√6/4 - √2/4 = (√6 - √2)/4

Difference

cos(120-45)

-1/2 ∗ √2/2 + √3/2 ∗ √2/2

-√2/4 + √6/4 = (-√2 + √6)/4 = (√6 - √2)/4

Answer 2

Since 2/3 -1/4 = 5/12, then 2π/3 - π/4 = 5π/12

and we know that cos(a - b) = cosa cosb + sina sinb

cos(2π/3) = -0.5

cos(π/4) = √2/2 ≈0.707

sin(2π/3) = √3/2 = 0.866

sin(√π/4) = √2/2 ≈ 0.707

Now plug in the numbers.