This course extends trigonometry from right triangles to angles of any size. It begins with the six ratios defined from a point, radian measure, signs in the four quadrants and the allied angle rule, so that any angle can be evaluated exactly. It then studies the domains, ranges and graphs of the trigonometric functions. The core of the course is the compound angle formulae and everything built on them: double, triple and half angles, exact values such as sin 18°, the sixty degree results, and the transformation formulae. It ends with identities in a triangle and other conditional identities, and with the periods of trigonometric functions.
Middle Years
Advanced Trigonometry
4-week course.
- 16 lessons
Overview
Lessons
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Trigonometric ratios and measuring angles
The six ratios defined from a point, the reciprocal, quotient and Pythagorean identities, and degrees, grades and radians with the radian values to know.
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Signs of trigonometric ratios and standard values
Which ratios are positive in each quadrant and why, and the table of values at 0°, 30°, 45°, 60° and 90°.
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Allied angles
Ratios of −θ, 90° ± θ, 180° ± θ, 270° ± θ and 360° − θ in terms of θ, and the short rule that produces any of them.
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Trigonometric values of any angle
Exact values for large angles and angles in radians, values on the axes, and where sine, cosine and tangent are zero.
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Domains, ranges and graphs of trigonometric functions
Domains and ranges of the six functions, their graphs and periods, and comparing values such as sin 1 and sin 1° from the graphs.
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Complementary angles and power identities
Results for complementary and supplementary angles, identities for sin⁴θ + cos⁴θ and sin⁶θ + cos⁶θ, least values from A.M. ≥ G.M., and the rules of proportion.
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Compound angle formulae
Sine, cosine, tangent and cotangent of A ± B, products into sums and back, three-angle formulae and the exact ratios of 15°.
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Identities from compound angles
Tangent and cotangent identities when A + B + C is 180° or 90°, the results for A + B = 45°, and evaluations built on them.
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Multiple and sub-multiple angles
Double, triple, half and third angle formulae, and the exact ratios of 18°, 36° and 22½°.
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Results from multiple angles
Exact values of tan 7½°, telescoping tangent chains, products of cosines of doubling angles, sums of cosines in A.P. and the angles 2π/7, 4π/7, 6π/7.
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Results with sixty degrees
Products and sums of ratios of θ, 60° + θ and 60° − θ, sums with 120° ± θ, and the results when two angles add to 60°.
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Square roots, signs and componendo and dividendo
Simplifying √(1 ± sin A) with the correct sign, the exact value of 4 sin 27°, and componendo and dividendo for ratios of sines and tangents.
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Transformation formulae
Products into sums and sums into products, the four identities for the angles of a triangle, and typical evaluations and proofs.
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Proving identities and evaluating products
Worked identities and exact products and sums of awkward angles, using conjugates, telescoping chains, supplements and the cosine product result.
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Conditional identities
Identities that hold under a condition: deriving one relation from another, identities in a triangle, and angle sums of 2S and 270°.
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Periodicity of trigonometric functions
Periodic functions and their periods, the rules for f(ax + b) and for sums, and the periods of powers and moduli of trigonometric functions.
Practice papers
Ten question papers that follow the lessons in order. Each has 20 questions for 50 marks (1 hour 30 minutes), with a key sheet and worked solutions.
MCQ tests
Ten tests of 25 multiple-choice questions (30 minutes each). Answer them on the page and check your score, or download the PDF with its key and solutions.
Class notes
Advanced Trigonometry
Typed from the teacher's handwritten class slides, 52 pages.
Download the notes (PDF, 2.2 MB)