Trigonometric equations
Objective. Find every solution in a stated interval.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Solving trigonometric equations within a stated interval requires finding the principal solution using inverse trig functions, then using the symmetry of the sine, cosine or tangent graph to find every other solution in range. For cos θ=0.5 with 0≤θ≤360°, the principal solution is θ=60°, and by symmetry the second solution is 360°−60°=300°. For sin θ=−0.5 the solutions come from a negative region, found using related angles and the correct quadrants. Always sketch the relevant graph or use the CAST diagram to ensure no valid solution in the interval is missed.
Worked example
Solve the trigonometric equation sin θ = √3/2 for 0° ≤ θ ≤ 360°.
- 1Find the principal solution: θ=60°
- 2Recognise sine is also positive in the second quadrant
- 3Compute the second solution as 180°−60°=120°
- 4State both solutions: θ=60° and θ=120°
Answer: θ = 60° or θ = 120°
Concept mastery
Based on your past attempts at this lesson — the question types to practise again come first.
Complete a practice set to see your concept breakdown here.
Practice
Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.
Start practisingCommon mistake. Students find only the calculator's principal value and stop, missing the second solution within the interval; for sin θ=0.5 on 0≤θ≤360°, both θ=30° and θ=150° must be given.
Your progress
Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.
0 of 5 steps complete
Your place saves automatically as you read.