Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — a basic equation, and finding the second solution
Solve 2sinθ=1 for 0°≤θ≤360°.
- 1sinθ=21Get the ratio alone on one side first.
- 2Principal value: θ=30°One of the angles worth memorising.
- 3sinθ is positive in the first and second quadrants.Ask which quadrants make the ratio positive — that tells you where the other solution is.
- 4Second solution: 180°−30°=150°For sine, the partner solution is 180°−θ.
- 5θ=30° and 150°Both lie inside the interval, so both count.
Example 2 — an equation that is quadratic in cosine
Solve 2cos2θ+cosθ−1=0 for 0°≤θ≤360°.
- 1Let c=cosθ, giving 2c2+c−1=0Substituting a letter makes the quadratic structure obvious.
- 2(2c−1)(c+1)=0, so c=21 or c=−1Both roots are between −1 and 1, so both are usable.
- 3cosθ=21: principal value 60°; cosine is also positive in the fourth quadrant, giving 360°−60°=300°For cosine the partner solution is 360°−θ.
- 4cosθ=−1: θ=180°Only one solution here — the cosine curve reaches −1 just once in a full turn.
- 5θ=60°,180°,300°Three solutions altogether. Stopping at two is the usual slip.
Example 3 — proving an identity
Prove that sinθcosθ1−cos2θ=tanθ.
- 1Start with the left-hand side — it is the messier one.Work towards the simpler side; never move terms across the equals sign.
- 21−cos2θ=sin2θThe Pythagorean identity, rearranged. This is almost always the first move.
- 3LHS =sinθcosθsin2θSubstitute it in.
- 4=cosθsinθCancel one sinθ from top and bottom — here it genuinely is a factor.
- 5=tanθ = RHSState that you have reached the right-hand side.