Solving Trig Equations and Inverse Trig Functions

Q: Why does my calculator give only one answer?

Inverse trig functions return a single principal value. The other solutions come from the symmetry of the unit circle and the fact that trig functions repeat.

Q: How many solutions does sin x = 0.5 have between 0° and 360°?

Two: 30° and 150°. Sine is positive in quadrants I and II.

Why $\sin x = \frac12$ has infinitely many solutions, what $\sin^{-1}$ , $\cos^{-1}$ and $\tan^{-1}$ return, and a reliable method for finding every solution in one turn using reference angles and quadrants.

3 min read Supports 3 lessons

$\sin x = \frac12$ at $x = 30°$ . But the sine wave repeats, and it also passes through $\frac12$ on the way down, at $150°$ . Then at $390°$ , $510°$ , and so on for ever. A trig equation usually has infinitely many solutions, so we normally ask for every solution in one turn, $0° \le x < 360°$ .

Within one turn, the wave crosses $y = \frac12$ twice.

Inverse trig functions

Your calculator's $\sin^{-1}$ undoes sine, but it can only return one answer, called the principal value.

Inverse trig functions

\sin^{-1}\!\left(\tfrac12\right) = 30^\circ, \quad \cos^{-1}\!\left(-\tfrac12\right) = 120^\circ, \quad \tan^{-1}(1) = 45^\circ

$\sin^{-1}$ returns an angle in $[-90°, 90°]$ , $\cos^{-1}$ in $[0°, 180°]$ and $\tan^{-1}$ in $(-90°, 90°)$ .

Watch out

$\sin^{-1}x$ means the inverse sine, not $\frac{1}{\sin x}$ .

Finding every solution

Use the unit circle. Each value of sine or cosine (other than the extremes) occurs at two points on the circle, mirror images of each other.

$\sin x = \frac12$ at $30°$ and at $150°$ , its mirror image across the $y$ -axis.

Where each function is positive

\sin > 0:\ \text{QI, QII} \qquad \cos > 0:\ \text{QI, QIV} \qquad \tan > 0:\ \text{QI, QIII}

$2\cos x + \sqrt3 = 0, \quad 0^\circ \le x < 360^\circ$
Isolate the trig function first.

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Tip

The method: isolate, find the reference angle, decide which quadrants the sign allows, then place the reference angle in each one.

Multiple angles

For $\sin 2x = \frac12$ , solve for $2x$ first over the doubled range $0° \le 2x < 720°$ , then halve. You get four solutions instead of two.

Key idea

Calculus finds turning points by solving $f'(x) = 0$ . For trig functions, that means solving equations exactly like these.

Common questions

Why does my calculator give only one answer?: Inverse trig functions return a single principal value. The other solutions come from the symmetry of the unit circle and the fact that trig functions repeat.
How many solutions does sin x = 0.5 have between 0° and 360°?: Two: $30°$ and $150°$ . Sine is positive in quadrants I and II.

Practise solving trig equations

The Solving Trig Equations lesson has worked examples and unlimited practice that adapts to you.

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Common questions

Why does my calculator give only one answer?

Inverse trig functions return a single principal value. The other solutions come from the symmetry of the unit circle and the fact that trig functions repeat.

How many solutions does sin x = 0.5 have between 0° and 360°?

Two: $30°$ and $150°$ . Sine is positive in quadrants I and II.