Find all real numbers in the interval [ 0 , 2pi ) that satisfy each equation. Round approximate answers to the nearest tenth.
1. tan(-x) = tan(x)
2. 2 cos^2(x) = 3 cos(x)
3. tan (x) = sec x - sqrt(3)
4. 5 sin^2(x) - 2 sin(x) = cos^2(x)
Thanks.
+128400 1. tan(-x) = tan(x)
Note that tan (-x) = -tan(x)....so we have
-tan (x) = tan(x) add tan (x) to both sides
0 = 2 tan x divide both sides by 2
0 = tan x .... and this occurs at x = 0 and x = pi
+128400 2. 2 cos^2(x) = 3 cos(x) subtract 3cos (x) from both sides
2cos^2(x) - 3 cos (x) = 0
cos (x) [ 2cosx - 3 ] = 0
So
cos x = 0.....and this occurs at x= pi/2 and x = 3pi/2
Or
2cosx - 3 = 0
2cosx = 3
cos x = 3/2 impossible
+128400 4. 5 sin^2(x) - 2 sin(x) = cos^2(x)
5sin^2 x - 2sin x = 1 - sin^2 x
6sin^2x - 2sin x - 1 = 0
(3sin x + 1) ( 2sin x - 1) = 0
3sin x = 1
sin x = 1/3 take the arcsin
arcsin (1/3) = x ≈ .34 rads and x ≈ [ pi - .34] rads ≈ 2.80 rads
And
2sin x - 1 = 0
2sin x = 1
sin x = 1/2 and this occurs at x = pi/6 and x = 5pi/6
