Evaluate giving exact values and identities. Calculator based solutions are not acceptable. Thanks for any help
cos (4pi/3 – 3pi/4)
Use the cos(A-B) identity.
4pi/3 – 3pi/4
= 7pi/12
you posted this version already, and I answered it.
To evaluate cos(4π/3 - 3π/4), we can use the sum-to-product identity for cosine. The sum-to-product identity states that:
cos(x - y) = cos(x)cos(y) + sin(x)sin(y)
Let's apply this identity to the given expression:
cos(4π/3 - 3π/4) = cos(4π/3)cos(3π/4) + sin(4π/3)sin(3π/4)
Now, let's determine the values of cos(4π/3) and sin(4π/3) using the unit circle. In the unit circle, the value of cosine is represented by the x-coordinate and the value of sine is represented by the y-coordinate.
For 4π/3, the reference angle is π/3. Since the terminal side for 4π/3 lies in the third quadrant, both the x-coordinate (cosine) and the y-coordinate (sine) will be negative.
From the unit circle:
cos(4π/3) = -1/2
sin(4π/3) = -√3/2
Next, let's find the values of cos(3π/4) and sin(3π/4) in the same way.
For 3π/4, the reference angle is π/4, and the terminal side falls in the second quadrant.
cos(3π/4) = -√2/2
sin(3π/4) = √2/2
Now, substitute these values into the equation:
cos(4π/3 - 3π/4) = (-1/2)(-√2/2) + (-√3/2)(√2/2)
Simplifying:
cos(4π/3 - 3π/4) = √2/4 + √6/4
Since the square root terms do not have any common factors, we can combine them by grouping:
cos(4π/3 - 3π/4) = (√2 + √6)/4
Therefore, the exact value of cos(4π/3 - 3π/4) is (√2 + √6)/4.