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Mathematical Methods
Circular functions
Chapter 9
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RESTS ON
Compute a circle's circumference and area from a stated radius or diameter
FOUNDATIONS
Substitute known values into a formula and evaluate it correctly
CH 1
Multiplication and division
CH 2
Read coordinates off a graphed line respecting its axis scales
CH 1
Signed numbers
CH 2
Solve a one-step linear equation
FOUNDATIONS
Radians, arc length and coterminal angles
Exact trigonometric ratios
Unit circle definitions
Exact values and symmetry
Basic sine and cosine graphs
Circular equations by bisection
Small-angle approximations
Convert an angle between degrees and radians, keeping an exact rational multiple of π where available and distinguishing a degree label from an unlabelled radian value
Use the defining relation between radius, central angle in radians and arc length in any direction, including minor and major arcs
Generate and recognise angles with the same terminal point by adding integer multiples of one full revolution in the active angle unit
Reduce an extended positive or negative rotation and identify its quadrant or axis boundary
Compute the length of an arc subtending θ° at the centre
Multiply and divide fractions
Simplify a fraction
State the exact trigonometric ratios from the named side lengths
Identify the hypotenuse and compute it from the two shorter sides
Compute a shorter side from the hypotenuse and the other leg
Name each side of a right-angled triangle relative to a marked angle
Use the terminal point (x, y) on the unit circle to obtain cos θ = x and sin θ = y, or recover the coordinate pair from known sine and cosine values
Obtain tan θ = y/x from a unit-circle point and recognise that tangent is undefined when x = 0
State and apply the maximal domains and ranges of the sine, cosine and tangent functions, including tangent's excluded odd half-turn boundaries
Evaluate sine, cosine or tangent at a stated degree or radian argument to a stated decimal precision, with the calculator mode matching the written angle unit
Rounding
Determine which of sine, cosine and tangent are positive, negative, zero or undefined in each quadrant or on a boundary
Reduce an angle to its first-quadrant reference angle measured from the horizontal axis
Obtain exact sine, cosine and tangent values for the standard first-quadrant angles, their symmetric points and axis boundaries
Apply reflection, parity and periodicity relations to rewrite a circular-function value in terms of a base angle and the correct sign
Find all angles in a stated finite domain whose sine, cosine or tangent has a given exact value, using reference angles and periodicity
Identify the domain shown, range, equilibrium, amplitude, period, intercepts and turning points of the basic sine and cosine functions, and relate their horizontal offset to complementary values
Construct and extend the quarter-period key-point table for a basic sine or cosine graph over a stated interval and count complete or partial cycles without double-counting shared endpoints
Connect intersections of a circular graph and an algebraic graph with solutions, and identify a root count or sign-change bracket from supplied graphical or tabular evidence
Apply a fixed number of bisection iterations to a continuous circular–algebraic difference function, retaining the half-interval whose endpoint values have opposite signs
Read the point of intersection of two graphed lines
Use and assess the prescribed local polynomial approximations for circular functions in radians, including substitution into a simple equation and comparison with an enclosed function value
Solve two-step linear equations by inverse operations in reverse order