About this topic
Before you begin
Explain signed trigonometric values and identities, interpret periodic models and solve equations on stated intervals in G3 Additional Mathematics.
Learning goals
- Produce all six exact signed trigonometric values
- Produce exact principal inverse trigonometric values
- Interpret and graph trigonometric functions
- Formulate, use and interpret a mathematical model
- Give the amplitude, period and range of a trigonometric graph from its equation
- Transform and justify trigonometric identities
- Simplify a trigonometric expression using the standard identities
- Produce complete exact trigonometric solution sets on stated intervals
- Express a cos x + b sin x in R form and find its greatest and least values
- Express a cos x + b sin x in R form and solve an equation in an interval
- 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
Choose a lesson
Work through them in order.
- Not startedUnit circle and inverse valuesNextUse signed unit-circle coordinates and exact ratios, then select principal inverse values without losing the original angle context.
- Not startedTrigonometric graphs and modelsNextRead input/output scaling, preserve tangent discontinuities and infer a bounded periodic model with correct phase units.
- Not startedTrigonometric identities and proofNextDerive exact values and identities, justify cancellations on the original domain and distinguish a proof from isolated checks.
- Not startedInterval equations and R formNextTrack transformed intervals, preserve zero factors and use signed R forms to solve equations and justify restricted ranges.