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Solving Trigonometric Equations

Solving a trig equation means finding every angle in a given interval that makes it true. Isolate the trig function first, then find all the matching angles.

Because sine and cosine repeat, an interval like 0∘≤θ<360∘0^\circ \leq \theta < 360^\circ usually has more than one solution. The reference angle and the quadrants give them all.

Isolate, then solve

Get the trig function alone, just like isolating a variable. 2sin⁡θ−1=02\sin\theta - 1 = 0 becomes sin⁡θ=12\sin\theta = \dfrac{1}{2}.

Then ask which angles have that value. The reference angle is the acute angle whose function matches the magnitude.

Find every solution in the interval

Sine is positive in quadrants I and II, so sin⁡θ=12\sin\theta = \dfrac{1}{2} gives θ=30∘\theta = 30^\circ and θ=150∘\theta = 150^\circ within one revolution.

Always check the sign to pick the right quadrants, and list every solution the interval allows.

The unit circle below shows why a positive sine has two answers: one angle in Quadrant I and one in Quadrant II reach the same height above the x-axis (the dashed line), so their sine values are equal.

Worked examples

Example 1: solving over one revolution

Solve 2sin⁡θ−1=02\sin\theta - 1 = 0 for 0∘≤θ<360∘0^\circ \leq \theta < 360^\circ.

Isolate sinesin⁡θ=12\sin\theta = \tfrac{1}{2}
Reference angle30∘30^\circ
Positive in QI and QII30∘,;150∘30^\circ, ; 150^\circ

Answer: θ=30∘\theta = 30^\circ and 150∘150^\circ

Example 2: isolating first

Rewrite 2cos⁡θ=12\cos\theta = 1 to isolate the trig function.

Divide by 2cos⁡θ=12\cos\theta = \tfrac{1}{2}

Answer: cos⁡θ=12\cos\theta = \dfrac{1}{2}

Example 3: a negative value picks other quadrants

Solve 2cos⁡θ+1=02\cos\theta + 1 = 0 for 0∘≤θ<360∘0^\circ \leq \theta < 360^\circ.

Isolate cosinecos⁡θ=−12\cos\theta = -\tfrac{1}{2}
Reference angle60∘60^\circ
Negative in QII and QIII120∘,  240∘120^\circ, \; 240^\circ

Answer: θ=120∘\theta = 120^\circ and 240∘240^\circ

Try one yourself

Common questions

Why is there more than one solution?

Sine and cosine repeat, so within a full revolution several angles can share the same value. List every one in the given interval.

What is a reference angle?

The acute angle whose trig value matches the magnitude of your equation. It locates the solutions in each quadrant.

How do I choose the quadrants?

By the sign of the isolated function. Positive sine means quadrants I and II; negative cosine means II and III, and so on.

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