Question

Find all exact angles in the interval [0,2pi] that satisfy the following equations. (a)cos x +cos 2x =0 (b) cos 2x=-0.5 (c)√(12) + 2 tan (x/2)=0 (c)cos 2x=-0.5 (d)sin (2x - (pi/2))= -0.5 ?

Answer 1

Solve: `sqrt12 + 2tan (x/2) = 0`
Ans: x = 240 deg

Explanation:

2tan (x/2) = - sqrt12
tan (x/2) = - sqrt12/2 = - 1.73
Calculator gives -> x/2 = -60 (or 300) and x/2 = -60 + 180 = 120 deg
`x/2 = 300` --> x = 600 = 240 + 360 -->`x = 240` deg

Check by calculator.
x = 240 --> x/2 = 120 --> 2tan 120 = -3.46 = sqrt12. OK

Answer 2

Solve `sin (2x - pi/2) = -0.5`

Ans: `(5pi)/6 and (7pi)/6`

Explanation:

`sin (2x - pi/2) = -0.5`
Trig table and trig unit circle give -->
`(2x - pi/2) = - pi/6` or `(11pi)/6` (same terminal).
The trig unit circle gives another `(2x - pi/2) = (7pi)/6` that has the same sine value (-0.5).

a. `2x - pi/2 = (7pi)/6` --> `2x = (7pi)/6 + pi/2 = (10pi)/6 = (5pi)/3`
--> `x = (5pi)/6`

b. `2x - pi/2 = (11pi)/6 --> 2x = (11pi)/6 + pi/2 = (14pi)/6 = (7pi)/3`
--> `x = (7pi)/6`