use a sum or difference formula to find the exact of the trigonometric function
cos(-13pi/12)
cos(-13pi/12)
= cos(-13pi/12 + 24pi/12)
= cos(11pi/12)
= -cos(pi - 11pi/12)
= -cos(pi/12)
now, cos(x/2) = √((1+cos(x))/2)
-cos(pi/12) = -√((1+cos(pi/6))/2)
= -√((1+√3/2)/2)
= -√(2 + √3/4)
= -1/2 √(2+√3)
or
-1/2√2 (1+√3)
To find the exact value of the trigonometric function cos(-13π/12), we can use the sum or difference formula for cosine. The sum or difference formula for cosine states:
cos(A ± B) = cos(A)cos(B) ∓ sin(A)sin(B)
In this case, let's express -13π/12 as the sum or difference of two angles. We can rewrite it as:
-13π/12 = -π/4 - 3π/3
Now, using the sum formula, we have:
cos(-13π/12) = cos(-π/4 - 3π/3)
Using the formula, we substitute A = -π/4 and B = 3π/3:
cos(-13π/12) = cos(-π/4)cos(3π/3) - sin(-π/4)sin(3π/3)
Now, let's evaluate the trigonometric functions for these angles:
cos(-π/4) = √2/2
cos(3π/3) = cos(π) = -1
sin(-π/4) = -√2/2
sin(3π/3) = sin(π) = 0
Substituting these values into our formula, we get:
cos(-13π/12) = (√2/2)(-1) - (-√2/2)(0)
Simplifying, we get:
cos(-13π/12) = -√2/2
Therefore, the exact value of cos(-13π/12) is -√2/2.