How many distinct solutions exist for the trigonometric equation 2\cos 2\theta = 1 within … — JAMB UTME

How many distinct solutions exist for the trigonometric equation $2\cos 2\theta = 1$ within the interval $0^\circ \le \theta \le 360^\circ$ ?

4 distinct solutions

The equation

\cos 2\theta = \frac{1}{2}

requires finding all angles whose cosine is

\frac{1}{2}

. Because the angle argument is

2\theta

, as

\theta

completes one full rotation (

0^\circ

360^\circ

2\theta

completes two full rotations (

0^\circ

720^\circ

). In two rotations, the cosine function takes the value

+\frac{1}{2}

exactly four times (twice per rotation), leading to four distinct solutions for

\theta

Isolate the trigonometric function in the equation

\cos 2\theta = \frac{1}{2}

Dividing both sides by 2 isolates the cosine term.

Determine the working interval for the multiple angle

2\theta

0^\circ \le 2\theta \le 720^\circ

Since

0^\circ \le \theta \le 360^\circ

, multiplying the inequality by 2 gives the domain for

2\theta

Find all values of

2\theta

where cosine is positive

\frac{1}{2}

in the domain

[0^\circ, 720^\circ]

2\theta = 60^\circ, 300^\circ, 420^\circ, 660^\circ

Cosine is positive in Quadrants I and IV. In the first revolution (

0^\circ \text{ to } 360^\circ

2\theta = 60^\circ

and

360^\circ - 60^\circ = 300^\circ

. Adding

360^\circ

for the second revolution gives

420^\circ

and

660^\circ

Solve for

\theta

by dividing each angle by 2

\theta = 30^\circ, 150^\circ, 210^\circ, 330^\circ

Dividing all four values of

2\theta

by 2 gives four distinct values of

\theta

within

[0^\circ, 360^\circ]

Solving trigonometric equations with multiple angles over a specified domain

Question