Trigonometric equations

Solve trig equations in a given domain: find the related angle, use ASTC to choose quadrants, handle boundary angles, compound angles and equations with two trig functions.

The method

Example 1●

Solve $\cos\theta = -\dfrac12$ for $0^\circ \le \theta \le 360^\circ$.

Related angle $60^\circ$.

$\theta = 180^\circ - 60^\circ$ or $180^\circ + 60^\circ$

cos is negative in quadrants 2 and 3.

$\theta = 120^\circ$ or $240^\circ$

Solutions are intersections

Each intersection of y = sin θ with the line y = k is a solution of sin θ = k.

Try this

Why are there no solutions when $k = 1.2$?
Compare the two solutions when $k = 0.5$. How are they related to $180^\circ$?

Boundary angles

When $k$ is $0$, $1$ or $-1$, the solutions are boundary angles. Read them from the unit circle or the graph, not from ASTC: $\sin\theta = -1$ gives only $\theta = 270^\circ$.

Compound angles

Example 2●●●

Solve $\sin 2\theta = \dfrac{\sqrt{3}}{2}$ for $0^\circ \le \theta \le 360^\circ$.

Domain for $2\theta$: $0^\circ \le 2\theta \le 720^\circ$.

If θ runs from 0° to 360°, then 2θ runs from 0° to 720°.

$2\theta = 60^\circ, 120^\circ, 420^\circ, 480^\circ$

Related angle 60°, sin positive in quadrants 1 and 2, then go round again.

$\theta = 30^\circ, 60^\circ, 210^\circ, 240^\circ$

More than one function

Example 3●●●

Solve $\sin\theta = 2\cos\theta$ for $0^\circ \le \theta \le 360^\circ$, to the nearest degree.

$\tan\theta = 2$

Divide both sides by cos θ (cos θ = 0 is not a solution, since then sin θ = ±1 ≠ 0).

Related angle $\tan^{-1} 2 \approx 63^\circ$. Tan is positive in quadrants 1 and 3.

$\theta \approx 63^\circ$ or $243^\circ$

Example 4: a quadratic in disguise●●●

Solve $2\sin^2\theta - \sin\theta - 1 = 0$ for $0^\circ \le \theta \le 360^\circ$.

$(2\sin\theta + 1)(\sin\theta - 1) = 0$

Let u = sin θ: 2u² − u − 1 = (2u + 1)(u − 1).

$\sin\theta = -\tfrac12$ or $\sin\theta = 1$

$\theta = 210^\circ, 330^\circ$ or $\theta = 90^\circ$

Flashcards

First step

Get one trig function alone.

Domain for 2θ

Double the given domain.

sin θ = cos θ type

Divide by cos θ to get tan θ.

sin θ = 1.2

No solution: sin is between −1 and 1.

Practice

Solve $\sin\theta = \dfrac{1}{\sqrt{2}}$ for $0^\circ \le \theta \le 360^\circ$. Enter both angles.
Solve $\tan\theta = -\dfrac{1}{\sqrt{3}}$ for $0^\circ \le \theta \le 360^\circ$.
Solve $\cos\theta = -1$ for $0^\circ \le \theta \le 360^\circ$.
How many solutions does $\cos 2\theta = \tfrac12$ have for $0^\circ \le \theta \le 360^\circ$?
Solve $\sin\theta = 0.3$ for $0^\circ \le \theta \le 360^\circ$, to the nearest degree.
$\sin\theta = \cos\theta$ for $0^\circ \le \theta \le 360^\circ$ gives:
$3\sin\theta + 5 = 0$ has how many solutions?
Solve $\sin\theta(2\cos\theta - 1) = 0$ for $0^\circ \le \theta \le 360^\circ$. Enter all five solutions.