Trigonometry : Solve the following problem

Question

Solve 2sinx= -1/2 for x ∈ [0,4π]

Answer 1

sinx = -1/4; x = π + sin^-1(1/4) + 2πn or x = 2π - sin^-1(1/4) + 2πn

.Let n = 0; x = π + sin^-1(1/4), x = 2π - sin^-1(1/4)

If n = 1; x = 3π + sin^-1(1/4), x = 4π - sin^-1(1/4).

This is all 4 solutions of given equation on interval [0, 4π]

Answer 2

2sinx = -1/2

sinx = -1/4, x is in the 3rd or 4th quadrants where sines <0

x = 360n +arcsin(-1/4) = about -14.4775 + 360n where n= any integer

or 194.4775 +360n where n = any integer

for the interval (0,4pi) = (0,180(4) degrees) = (0,720)

x = about 360-14.4775, 194.4775, 360+194.4775, 360+345.4775

= about 345.5, 194.5, 554.5, 705.5

4 solutions

Answer 3

Answer 4

sinx = -1/4

Sinx is negative in quadrants 3 and 4.

Reference angle = Sin^-1(1/4) = 0.25268

Solutions in [0,2π] are 0.25268 + π and 2π - 0.25268

To get the solutions in (2π, 4π], add 2π to each of the solutions above.

We obtain 0.25268 + 3π and 4π - 0.25268.

So, there are a total of 4 solutions in {0, 4π].