Trigonometry and modelling
Combine angles, rewrite trigonometric expressions and model repeating changes.
Learn compound, double, half and triple angle formulae, solve equations and combine waves with the R formula. Further lessons on trigonometric methods and modelling are being added.
- Compound angle formulaeDerive and use sine, cosine and tangent addition formulae. Explore signed components, expand compound angles and practise with worked solutions.
- Exact values using compound anglesFind exact trigonometric values with compound angle formulae, signed triangles and quadrant checks. Apply them to line angles, triangle geometry and algebraic relationships.
- Double angle formulaeDerive sin 2x, cos 2x and tan 2x, choose the right cosine form and calculate exact values. Explore angle doubling, circle geometry and parameter elimination.
- Half angle formulaeUse half angle formulae to find exact sine, cosine and tangent values. Choose signs from the half-angle quadrant and apply the identities to bisectors and triangle geometry.
- Compound and double angle equationsSolve compound and double angle equations by expansion, factorisation and quadratic substitution. Keep zero roots, check tangent domains and find every answer in the interval.
- Tangent half angle substitutionUse t = tan(θ/2) to turn trigonometric equations into rational algebra. Derive the formulae, recover angle branches and check missing odd multiples of π.
- Triple angle formulaeDerive the triple angle formulae for sine, cosine and tangent. Solve multiple-angle equations, connect trigonometry with cubics and keep every interval solution.
- R formula for trigonometric expressionsWrite a sin x + b cos x as one sine or cosine using the R formula. Match coefficients, choose the correct phase quadrant and understand amplitude and graph shifts.
- Solving equations with the R formulaSolve a sin x + b cos x = c using the R formula. Transform the angle interval, find every branch, handle multiple angles and reject excluded or squared candidates.
- Trigonometric maxima and minimaFind trigonometric maxima, minima and ranges using amplitude, double angles and completing the square. Check restricted intervals, reciprocal signs and whether bounds are attained.