Question

How do you evaluate `sec110°`?

Answer 1

Find `sec 110`

Ans: -2.94

Explanation:

`sec 110 = 1/(cos 110)`. Find cos 110 without using calculator.
Call cos (110) = cos x
Use the formula: `cos 3x = 4cos^3 x - 3cos x.` We get:
cos 330 = cos (-30 + 360) = cos 30 = sqrt3/2
`cos 330 = sqrt3/2 = 4cos^3 x - 3cos x`
`4cos^3 x - 3cos x - sqrt3/2 = 0`
Solve this cubic equation by graphing calculator
By estimation, we get;
cos x = - 0.34 --> x = 110 (accepted) --> 3x = 330
cos x = - 0.65 --> x = 130 (rejected) --> 3x = 390
cos x = 0.98 --> x = 10 (rejected) --> 3x = 30
Finally:
`sec (110) = 1/(-0.34) = - 2.94`
graph{4x^3 - 3x - sqrt3/2 [-2.5, 2.5, -1.25, 1.25]}