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 ?

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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 ?

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Answer 1 · 2017-12-14T17:24:19

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 $x = 63.43 + 180 = 243^@43$

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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

Explanation:

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