1) Use the cofunction identites to evaluate the expression without the aid of a calculator.
cos^2 18 degrees + cos^2 72 degrees.
Would I use the forumula
cos(pi/2 - Q) = sin Q
see previous post
Yes, you can use the cofunction identities to evaluate the expression without a calculator. In this case, you can use the identity:
cos²(θ) = sin²(π/2 - θ).
Applying this identity, you can rewrite cos²(18°) as sin²(π/2 - 18°) and cos²(72°) as sin²(π/2 - 72°).
Now we can evaluate each term separately:
For cos²(18°), use the identity sin²(π/2 - θ):
sin²(π/2 - 18°).
To evaluate this expression without a calculator, we need to recognize the angle 18° as a special angle.
Since sin(30°) = 1/2, let's express 18° as a sum or difference of angles involving 30°:
18° = 30° - 12°.
Now, we can use the identity sin(A - B) = sin(A)cos(B) - cos(A)sin(B).
sin(30° - 12°) = sin(30°)cos(12°) - cos(30°)sin(12°).
Since sin(30°) = 1/2 and cos(30°) = √3/2, we can substitute these values into the formula:
sin(30° - 12°) = (1/2)cos(12°) - (√3/2)sin(12°).
Now, we need to evaluate cos(12°) and sin(12°).
Using the cofunction identity cos(90° - θ) = sin(θ), we can rewrite cos(12°) as sin(90° - 12°).
cos(12°) = sin(90° - 12°).
Now, we can evaluate sin(12°) without a calculator by recognizing it as a special angle.
sin(12°) = sin(30° - 18°).
Using the same formula as before, sin(30° - 18°) = (1/2)cos(18°) - (√3/2)sin(18°).
To evaluate sin(18°) without a calculator, we can use another special angle.
sin(18°) = sin(30° - 12°).
sin(30° - 12°) = (1/2)cos(12°) - (√3/2)sin(12°).
Now, use these values to find sin²(π/2 - 18°).
Finally, apply the same process to find sin²(π/2 - 72°).
Once you have calculated sin²(π/2 - 18°) and sin²(π/2 - 72°), sum them together to get the final result.