How do you find the exact value of cos -7pi/12 without using a calculator to change radian to degree
use a unit circle
To find the exact value of cos(-7π/12) without using a calculator, you can use a half-angle formula and other trigonometric identities.
1. Start by recognizing that -7π/12 is an odd multiple of π/12. This means that cos(-7π/12) is related to cos(π/12), which is a fraction we can compute.
2. To calculate cos(π/12), we can use half-angle formulas. The half-angle formula for cosine is cos(x/2) = ± √((1 + cos(x))/2).
3. Apply the half-angle formula to cos(π/6) = 2cos^2(π/12) - 1. Rearrange the equation:
2cos^2(π/12) = cos(π/6) + 1
4. Plug in the value of cos(π/6), which is √3/2:
2cos^2(π/12) = √3/2 + 1
5. Simplify the equation by multiplying both sides by 2:
4cos^2(π/12) = √3 + 2
6. Solve for cos^2(π/12):
cos^2(π/12) = (√3 + 2)/4
7. Finally, take the square root of cos^2(π/12) to find cos(π/12):
cos(π/12) = ± √((√3 + 2)/4)
Note: The ± in the equation represents the fact that cos(π/12) can have two possible values, depending on the quadrant it lies in.