This course builds trigonometry from the idea of an angle as a rotation. You measure angles in degrees, grades and radians and convert between them, then define the six trigonometric ratios for any angle and learn the three fundamental identities. You find the signs of the ratios in each quadrant, the values at 0°, 30°, 45°, 60° and 90°, and the ratios of allied angles such as 180° − θ and 270° + θ. Next you study the ratios as functions: their domains, ranges, graphs and where they increase or decrease. The course ends with problem-solving ideas: least values, square roots, sums over complementary angles, and eliminating θ.
Middle Years
Trigonometric Ratios and Applications
Angles in degrees, grades and radians, the six trigonometric ratios, identities, signs in the quadrants, standard and allied angles, domains, ranges and graphs, and problem-solving concepts.
- 10 lessons
Overview
Lessons
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Angles and their measurement
An angle as a rotation, positive and negative angles, the sexagesimal, centesimal and radian systems, conversions between them, and the arc length l = rθ.
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The trigonometric ratios
The six ratios defined from a point on the terminal side, the right-triangle form for acute angles, reciprocal and quotient relations, and finding all ratios from one.
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Trigonometric identities
The identities sin²θ + cos²θ = 1, sec²θ − tan²θ = 1 and cosec²θ − cot²θ = 1, their useful forms, higher powers of sine and cosine, and proving identities.
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Signs of the trigonometric ratios
Which ratios are positive in each quadrant, the All Silver Tea Cups rule, and finding all six ratios of an angle from one ratio and its quadrant.
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Ratios of standard angles
Deriving the ratios of 0°, 30°, 45°, 60° and 90° from simple figures, the table of values, and evaluating expressions and finding angles with it.
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Allied angles
Ratios of −θ, 90° ± θ, 180° ± θ, 270° ± θ and n·360° ± θ, the rule for reducing any angle to an acute one, and where the ratios are zero.
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Domain, range and graphs
The domain, range and period of each ratio, the graphs of sin, cos, tan, cot, sec and cosec, and comparing values such as sin 1, sin 2 and sin 3.
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Increasing and decreasing ratios
Increasing and decreasing functions, the intervals where sine, cosine and tangent rise and fall, and comparing two values of the same ratio.
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Conceptual problems: least values and angle sums
Least values from AM ≥ GM, square roots as absolute values, and sums that collapse over complementary, supplementary and related angles.
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Conceptual problems: eliminating θ
Ratio and proportion rules, the product hexagon, identities to recognise, eliminating θ from pairs of equations, and cross-multiplication.