Trigonometric equations
Find every solution in an interval by considering signs and symmetry.
What you will learn
- Find a reference angle.
- Select all quadrants consistent with the equation.
Isolate the trigonometric ratio
Rearrange the equation until a single sine, cosine or tangent is isolated. Find a reference angle using an exact value or an inverse function. An inverse calculator result gives one value, not automatically every solution in the requested interval.
Sine is positive in quadrants I and II, cosine in I and IV, and tangent in I and III. Use the sign together with the reference angle, then check the original equation.
Respect the interval and units
A solution outside the stated interval must be excluded. Endpoints need particular care because an inclusive interval may contain both 0° and 360°. Check whether the question uses degrees or radians before forming solutions.
Worked example
Solve 2cos θ=−1 for 0°≤θ≤360°.
Show the worked solution
- Rearrange to cos θ=−1/2, with reference angle 60°.
- Cosine is negative in quadrants II and III.
- The angles are 180°−60° and 180°+60°.
Answer 120° and 240°
Common mistakes
- One inverse-trig output is not a complete solution set.
- Do not mix radian and degree values.
What can you explain now?
Solve sin θ=1/2 for 0≤θ≤2π.
Compare with the explanation
π/6 and 5π/6
The positive sine solutions lie in quadrants I and II.
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These lessons teach the topics represented in our current practice sets. They are not a complete course for every paper or option in the qualification.
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