Question

Find the value of θ from the following: 

2 sin 2θ = 1

Answer 1

1. Isolate the sine function

First, divide both sides of the equation by 2 to isolate the trigonometric term:

2 sin 2θ = 1

`sin 2θ = 1/2`

2. Find the reference angle

Determine which angle has a sine value of `1/2`.

From the standard unit circle values:

`sin (30^circ) = 1/2` or `sin (π/6) = 1/2`

Therefore, the argument 2θ can be:

2θ = 30° + 360°n or 2θ = 150° + 360°n

Where represents the number of full rotations.

3. Solve for θ

Divide the entire expression by 2 to find the value of θ:

Case 1: 2θ = 30°

⇒ θ = 15°

Case 2: 2θ = 150°

⇒ θ = 75°

In radians, these values correspond to:

`θ = π/12 + nπ` and `θ = (5π)/12 + nπ`

The values of θ that satisfy the equation are θ = 15° and θ = 75° for the first rotation.