Question

Find the exact value of Cos 7pie/12

Answer 1

The value is ((square root 2)- (square root 6))/4

Explanation:

We can find the value by breaking up the cos (7pi/12) into

cos (4pi/12 + 3pi/12)

This is the same thing as

cos (pi/3 + pi/4)

You can then rewrite that as

cos(pi/3) * cos(pi/4) - sin(pi/3) * sin(pi/4)

Then solve for each value

1/2 * ((square root 2)/2) - ((square root 3)/2) * ((square root 2)/2)

Multiply each value that should be together

((square root 2)/4) - ((square root 6)/4)

Common denominators allow the answer to be further simplified to

((square root 2)- (square root 6))/4

Answer 2

The exact value of cos(7π/12) can be found using trigonometric identities, specifically the cosine sum formula, resulting in -(√6 + √2)/4.

The exact value of cos(7π/12) is not directly found on the unit circle, so we need to use trigonometric identities to break it down into angles we know. The angle can be expressed as the sum or difference of angles that are easier to work with, like cos(7π/12) = cos(3π/4 + π/6). Now, we apply the cosine sum formula: cos(A + B) = cos(A)cos(B) - sin(A)sin(B).

For our angles, cos(3π/4) = -√2/2, sin(3π/4) = √2/2, cos(π/6) = √3/2, and sin(π/6) = 1/2. This gives us:
cos(7π/12) = (-√2/2)(√3/2) - (√2/2)(1/2) = -√6/4 - √2/4 = -(√6 + √2)/4, which is the exact value.