The exact value of cos(23π/12) is 0.7071067811865475, not 0.9365234037742956.
Method 1: Cosine Sum Identity
Express 23π/12 as the sum of two angles:
23π/12 = 18π/12 + 5π/12 = 3π/2 + π/4
Apply the cosine sum identity:
cos(a + b) = cos a cos b - sin a sin b
Substitute the angles:
cos(23π/12) = cos(3π/2 + π/4) = cos(3π/2) cos(π/4) - sin(3π/2) sin(π/4)
Use the trigonometric table values:
cos(3π/2) = 0, cos(π/4) = √2/2, sin(3π/2) = -1, sin(π/4) = √2/2
Calculate the result:
cos(23π/12) = (0)(√2/2) - (-1)(√2/2) = √2/2 = 0.7071067811865475
Method 2: Cosine Difference Identity
Express 23π/12 as the difference of two angles:
23π/12 = 2π + π/12
Apply the cosine difference identity:
cos(a - b) = cos a cos b + sin a sin b
Substitute the angles:
cos(23π/12) = cos(2π - π/12) = cos(2π) cos(π/12) + sin(2π) sin(π/12)
Use the trigonometric table values:
cos(2π) = 1, cos(π/12) = √2/2, sin(2π) = 0, sin(π/12) = √2/2
Calculate the result:
cos(23π/12) = (1)(√2/2) + (0)(√2/2) = √2/2 = 0.7071067811865475
Therefore, the exact value of cos(23π/12) is 0.7071067811865475, not 0.9365234037742956.
Question
How do you find the exact functional value cos 23pi/12 using the cosine sum or difference identity?