Question

Find all exact angles,x in the interval[-pi, pi] that satisfy (a)2cos x=√3 (b)√3 sin x = cos x (c) 4sin²x = √(12) sin x (d) sec² x=2 ?

Answer 1

a. `x = +- pi/6`
b. `x = pi/6, and x = (7pi)/6`
c. `x = pi/3, and x = (2pi)/3`
d. `63^@43, and x = 243^@43`

Explanation:

(a). `2cos x = sqrt3` --> `cos x = sqrt3/2`
Trig table and unit circle gives 2 solutions:
`x= +- pi/6`, or `x = +- 30^@`
(b). `sqrt3sin x = cos x`
Divide both sides by cos x (condition `cos x != 0`)
`sqrt3tan x = 1`--> `tan x = 1/sqrt3 = sqrt3/3`
Trig table and unit circle give 2 solutions:
`x = pi/6` and `x = pi/6 + pi = (7pi)/6`
(c). `4sin ^2 x = sqr12sin x = 2sqrt3sin x`.
Simplify by sin x (condition `sin x != 0`).
`2sin x = sqrt3` --> `sin x = sqrt3/2`
Trig table and unit circle -->
`x = pi/3`, and `x = (2pi)/3`
(d). `sin x.sec x = 2`.
`(sin x)(1/cos x) = tan x = 2`
Calculator and unit circle give 2 solutions:
`x = 63^@43`, and `x = 63.43 + 180 = 243^@43`