Question

How do you solve `sec^2x+2tanx=6` in the interval [0,360]?

Answer 1

`45^@, 90^@`,

Explanation:

`sec^2 x + 2tan x = 6`
`1/(cos^2 x) + (2sin x)/(cos x) = 6`
`1 + 2sin x.cos x = 6cos^2 x`
Use trig identities:
sin 2x = 2sin x.cos x
`2cos^2 x = 1 + cos 2x`
In this case we have:
`1 + 2sin x = 3 + 3cos 2x`
`2sin 2x - 3cos 2x = 2`
`sin 2x - (3/2)cos 2x = 1`
Call `tan t = sin t/(cos t) = 3/2 = tan 56^@31`, we get:
`sin 2x.cos t - sin t.cos 2x = cos 56^@31 = 0.55`
sin (2x - 56.31) = 0.55
Use calculator and unit circle -->
a. `(2x - 56.31) = 33^@68`
2x = 33.68 + 56.32 = 90^@ --> `x = 45@`
b. (2x - 56.31) = 180 - 56.31 = 123^@69
`2x = 123.69 + 56.31 = 180^@ -->`x = 90^@#