Trigonometric Equations
Trigonometric equations are mathematical statements that require finding all angles where a trigonometric function equals a given value. The equation sin x = 1/2, for example, has solutions at 30° and 150° within one complete rotation. These equations appear throughout GCSE Higher and A-level mathematics, requiring systematic approaches to find all solutions within specified intervals.
Why it matters
Trigonometric equations model periodic phenomena across engineering and physics. Radio waves, alternating current, and sound frequencies all follow trigonometric patterns where specific values occur at predictable intervals. In structural engineering, architects solve trigonometric equations to calculate roof angles and load distributions. The equation sin(ωt) = 0.707 determines when an AC voltage reaches 70.7% of its peak value, occurring twice per cycle at approximately 45° intervals. Weather forecasting uses trigonometric equations to model seasonal temperature variations and tidal patterns. GPS navigation systems rely on solving complex trigonometric equations involving satellite positions and signal timing. These applications require finding all solutions within relevant time periods or angular ranges, making the systematic solution methods essential for real-world problem solving.
How to solve trigonometric equations
Trig Equations
- Isolate the trig function: e.g. sin x = v.
- Find the reference angle from the unit circle.
- Use ASTC to list all solutions in the required interval [0, 2π) or [0°, 360°).
- For sin(kx) = v, solve for kx first, then divide. Remember the period.
Example: 2 sin x = 1 → sin x = 12 → x = π/6 or 5π/6 in [0, 2π).
Worked examples
Solve tan(x) = √3 on the interval [0°, 360°].
Answer: 60°, 240°
- Identify the reference angle from the unit circle → tan(reference) = √3 — Start with the positive version of the value and find the acute angle whose sin/cos/tan equals it. That's the reference angle.
- Find every angle in [0°, 360°] with the correct sign → x ∈ {60°, 240°} — Use ASTC to determine which quadrants give the desired sign. Each quadrant gives one solution (or two for the axial angles 0°, 90°, 180°, 270°, 360°).
Solve tan(x) = −1 on the interval [0, 2π].
Answer: 3π/4, 7π/4
- Find the reference angle in radians → reference angle from unit circle — The standard reference values in radians are π/6, π/4, π/3, π/2. Pick the one whose sin/cos/tan matches the absolute value of the right-hand side.
- List every solution in [0, 2π] → x ∈ {3π/4, 7π/4} — Apply ASTC to pick the right quadrants, then convert each to its radian form.
Solve cos(2x) = −√22 on the interval [0, 2π].
Answer: 3π/8, 5π/8, 11π/8, 13π/8
- Substitute u = 2x and find the new interval for u → u ∈ [0, 4π] — Since x ∈ [0, 2π] and u = 2x, the interval for u is [0, 4π] — 2 times longer, so expect 2× as many solutions as the standard equation.
- Solve cos(u) = −√22 and divide each solution by 2 → x ∈ {3π/8, 5π/8, 11π/8, 13π/8} — Find the base solutions, add 2π each time to stay in the longer interval, then divide by the coefficient.
Common mistakes
- Finding only one solution when multiple exist: solving sin x = 1/2 and giving x = 30° instead of x = 30°, 150° within [0°, 360°].
- Incorrect period handling in multi-angle equations: solving sin(2x) = 1/2 over [0, 2π] but missing solutions because the period doubles to 4π.
- Sign errors with ASTC quadrants: solving cos x = -1/2 and incorrectly placing solutions in quadrants I and II instead of quadrants II and III.
- Forgetting to divide by the coefficient: solving cos(3x) = 1/2 and leaving answers as 3x = π/3, 5π/3 instead of x = π/9, 5π/9.