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 ?

1 Answer
Dec 14, 2017

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#