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

I need to solve 2sin(x) = -1/2 for x ∈ [0, 4π].

Step 1: Isolate sin(x)

Divide both sides by 2: sin⁡(x)=−14sin(x)=−41​

Step 2: Find the reference angle

I need to find where sin(x) = -1/4. Let me find the reference angle: α=arcsin⁡(14)≈0.2527α=arcsin(41​)≈0.2527 radians (or about 14.48°)

Step 3: Determine which quadrants

Since sin(x) is negative, x must be in:

1. Quadrant III or Quadrant IV

Step 4: Find solutions in [0, 2π]

For one period [0, 2π]:

1. Quadrant III: $x_1 = \pi + \alpha = \pi + \arcsin\left(\frac{1}{4}\right) \approx 3.394$ radians
2. Quadrant IV: $x_2 = 2\pi - \alpha = 2\pi - \arcsin\left(\frac{1}{4}\right) \approx 6.030$ radians

Step 5: Find all solutions in [0, 4π]

Since the sine function has period 2π, I need to add 2π to each solution:

First period [0, 2π]:

1. $x_1 = \pi + \arcsin\left(\frac{1}{4}\right)$
2. $x_2 = 2\pi - \arcsin\left(\frac{1}{4}\right)$

Second period [2π, 4π]:

1. $x_3 = \pi + \arcsin\left(\frac{1}{4}\right) + 2\pi = 3\pi + \arcsin\left(\frac{1}{4}\right)$
2. $x_4 = 2\pi - \arcsin\left(\frac{1}{4}\right) + 2\pi = 4\pi - \arcsin\left(\frac{1}{4}\right)$

Final Answer

The four solutions are: x=π+arcsin⁡(14),2π−arcsin⁡(14),3π+arcsin⁡(14),4π−arcsin⁡(14)x=π+arcsin(41​),2π−arcsin(41​),3π+arcsin(41​),4π−arcsin(41​)

Or approximately: x≈3.394,6.030,9.678,12.313 radiansx≈

Answer 3

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 4

Answer 5

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].