Trigonometry
Build and connect the representations and methods in trigonometry.
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Before you begin
Build and connect the representations and methods in trigonometry.
Learning goals
- Interpret and graph trigonometric functions
- Formulate, use and interpret a mathematical model
- Transform and justify trigonometric identities
- Solve trigonometric equations within stated intervals
Syllabus statements covered
- Six trigonometric functions for angles of any magnitude (in degrees or radians) • Principal values of sin⁻¹x, cos⁻¹x, tan⁻¹x • Exact values of the trigonometric functions for special angles (30°, 45°, 60°) or (π/6, π/4, π/3) • Amplitude, periodicity and symmetries related to sine and cosine functions • Graphs of y = a sin(bx) + c, y = a sin(x/b) + c, y = a cos(bx) + c, y = a cos(x/b) + c and y = a tan(bx), where a is real, b is a positive integer and c is an integer • Use of: sin A/cos A = tan A; cos A/sin A = cot A; sin² A + cos² A = 1; sec² A = 1 + tan² A; cosec² A = 1 + cot² A; expansions of sin(A ± B), cos(A ± B), tan(A ± B); formulae for sin 2A, cos 2A, tan 2A; a cos θ + b sin θ in the form R cos(θ ± α) or R sin(θ ± α) • Simplification of trigonometric expressions • Solution of simple trigonometric equations in a given interval (excluding general solution) • Proofs of simple trigonometric identities • Use of trigonometric functions as models
- Six trigonometric functions for angles of any magnitude (in degrees or radians)
- Principal values of sin⁻¹x, cos⁻¹x, tan⁻¹x
- Exact values of the trigonometric functions for special angles (30°, 45°, 60°) or (π/6, π/4, π/3)
- Amplitude, periodicity and symmetries related to sine and cosine functions
- Graphs of y = a sin(bx) + c, y = a sin(x/b) + c, y = a cos(bx) + c, y = a cos(x/b) + c and y = a tan(bx), where a is real, b is a positive integer and c is an integer
- Use of: sin A/cos A = tan A; cos A/sin A = cot A; sin² A + cos² A = 1; sec² A = 1 + tan² A; cosec² A = 1 + cot² A; expansions of sin(A ± B), cos(A ± B), tan(A ± B); formulae for sin 2A, cos 2A, tan 2A; a cos θ + b sin θ in the form R cos(θ ± α) or R sin(θ ± α)
- Simplification of trigonometric expressions
- Solution of simple trigonometric equations in a given interval (excluding general solution)
- Proofs of simple trigonometric identities
- Use of trigonometric functions as models
Lessons
Work through them in order.
About lesson progress
Lesson marks record what you have studied. Checks show understanding separately. Guest marks come from this device.
- Unit circle and inverse valuesUse signed unit-circle coordinates and exact ratios, then select principal inverse values without losing the original angle context.
- Trigonometric graphs and modelsRead input/output scaling, preserve tangent discontinuities and infer a bounded periodic model with correct phase units.
- Trigonometric identities and proofDerive exact values and identities, justify cancellations on the original domain and distinguish a proof from isolated checks.
- Interval equations and R formTrack transformed intervals, preserve zero factors and use signed R forms to solve equations and justify restricted ranges.