Overview

Solve trig equations by isolating a trig function, finding reference angles, then checking all quadrants.

Core Skills

Use Reference Angles

Find the acute reference angle, then place solutions in the correct quadrants for the given sign.

Solve Quadratic Forms

If the equation is quadratic in $\sin\theta$ or $\cos\theta$, substitute a variable, solve the quadratic, then convert back.

Check the Interval

List all solutions in the requested interval (often $[0,360^\circ)$).

Basic Equation Template

If $\sin\theta = a$ with $|a|\le 1$, then

• $\theta=\theta_0$ and $\theta=180^\circ-\theta_0$ in $[0,360^\circ)$ when $a>0$.
• Use quadrants III/IV when $a<0$.

Quadratic Trig Equations

If an equation is quadratic in $\sin\theta$ or $\cos\theta$, substitute $u=\sin\theta$ or $u=\cos\theta$, solve, then convert to angles.

Worked Example

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

Let $u=\cos\theta$. Then $(2u+1)(u-1)=0$, so $u=1$ or $u=-1/2$. Thus $\theta=0^\circ$ and $\theta=120^\circ,240^\circ$.

More Examples

Example 1: Sine Equation

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

$\theta=60^\circ,120^\circ$.

Example 2: Tangent Equation

Solve $\tan\theta=1$ for $0^\circ\le\theta<360^\circ$.

$\theta=45^\circ,225^\circ$.

Example 3: Quadratic in Sine

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

$(2u-1)(u+1)=0$ gives $u=1/2$ or $u=-1$. So $\theta=30^\circ,150^\circ,270^\circ$.

Strategy Checklist

• Isolate the trig function first.
• Use substitution for quadratic forms.
• Enumerate all solutions in the given interval.

Common Pitfalls

• Forgetting all solutions in $[0,360^\circ)$.
• Missing the negative root in quadratic substitutions.
• Dropping angles with the same reference angle.

Practice Problems