Circular functions: angles, cycles and complete solutions: VCE Mathematical Methods Units 3 & 4

Loav · VCE Mathematical Methods · Units 3 & 4

F05 · 2026 curriculum · Original teaching guide

1. Radians connect angle to the unit circle

One complete turn is 2π radians or 360 degrees. Therefore π radians equals 180 degrees. Convert degrees to radians by multiplying by π/180; convert radians to degrees by multiplying by 180/π. In Methods function work, radians are the standard angle measure unless degrees are explicitly specified. A calculator set to the wrong mode can produce a plausible-looking but completely wrong answer.

On the unit circle, the point reached by angle θ from the positive x-axis has coordinates (cos θ, sin θ). This definition extends sine and cosine beyond acute angles and beyond a single rotation. Cosine is the horizontal coordinate; sine is the vertical coordinate. Positive angles rotate anticlockwise. The signs in each quadrant follow from the coordinate signs rather than from a separate memorised rule.

The special angles π/6, π/4 and π/3 correspond to 30, 45 and 60 degrees. Their sine values are 1/2, √2/2 and √3/2; their cosine values occur in the reverse order. Use a reference angle to find the magnitude, then the quadrant to determine the sign. For example, sin(7π/6) = −1/2 because 7π/6 is in the third quadrant with reference angle π/6.

Tangent is the ratio sin θ/cos θ wherever cos θ is non-zero. It is therefore undefined at θ = π/2 + nπ for integer n. The unit-circle identity sin² θ + cos² θ = 1 comes from the equation of the circle. This identity can simplify equations, but a squared value does not determine the sign of an unsquared sine or cosine without quadrant or domain information.

2. Periodicity organises the graph

Sine and cosine repeat every 2π. Both have domain ℝ and range [−1, 1]. Sine starts at zero, reaches 1 at π/2, returns to zero at π, reaches −1 at 3π/2 and returns to zero at 2π. Cosine starts at 1 and follows the same cycle shifted horizontally. These quarter-period anchors are more reliable for sketching than plotting many approximate points.

Tangent repeats every π, has range ℝ, and has a separate continuous branch between each pair of neighbouring vertical asymptotes. Its zeros occur at nπ and its asymptotes at π/2 + nπ. Do not join branches across an asymptote. Unlike sine and cosine, tangent has no maximum, minimum or amplitude. A vertical scale factor changes its outputs but does not create an amplitude.

3. Transform the angle before reading the phase shift

For y = A sin(B(x − C)) + D or the corresponding cosine rule, with A and B non-zero, the amplitude is |A|, period is 2π/|B|, and midline is y = D. The full-domain range is [D − |A|, D + |A|]. A negative A reflects the graph vertically. The horizontal shift C is visible only after factoring B from the entire angle expression.

For transformed tangent, the period is π/|B|. Find asymptotes by solving B(x − C) = π/2 + nπ, and find the midline crossings by solving B(x − C) = nπ. This direct angle equation is safer than trying to memorise a separate formula for every shifted graph. The resulting input restrictions belong to the domain.

4. Find every angle, not only the calculator’s answer

A circular equation can have several solutions in a bounded interval and infinitely many over ℝ. An inverse-trigonometric calculator button returns a principal value, not every solution. Use that value as a reference and then apply symmetry and periodicity. For sin u = sin α, solutions are u = α + 2nπ or u = π − α + 2nπ. For cos u = cos α, they are u = ±α + 2nπ. For tan u = tan α, they are u = α + nπ, with integer n.

In radians, the principal value from inverse sine lies in [−π/2, π/2], inverse cosine in [0, π], and inverse tangent in (−π/2, π/2). These are the angle ranges returned by the inverse operations; they are not the complete solution sets of the original periodic equations. Inverse tangent cannot return either endpoint because tangent is undefined there.

For a non-special-angle example, solve sin x = 3/10 on 0 ≤ x < 2π. Define α = sin⁻¹(3/10), where α ≈ 0.304693 radians. The positive sine occurs in the first and second quadrants, so the complete interval solution is x = α or x = π − α, approximately 0.305 and 2.837 radians. Over all real x, the families are α + 2nπ and π − α + 2nπ for integer n. Keep α in exact inverse-function form until a decimal is requested.

When the angle is a transformed expression, first translate the original x-interval into an interval for that angle. Solve within the angle interval, then undo the transformation. This approach makes the number of cycles explicit and reduces the chance of losing a solution. If the transformation has a negative coefficient, reorder the resulting interval bounds after solving.

5. Algebra can reveal a circular equation’s structure

An equation involving sin² x and sin x can be treated as a quadratic in u = sin x. After solving for u, retain only values in [−1, 1], then solve the corresponding circular equations on the original interval. For example, a candidate u = 2 is rejected because no real angle has sine 2. This range check is analogous to requiring a positive substituted value when u = eˣ.

Do not divide an equation by sin x or cos x without considering when that divisor is zero. For sin x(2 cos x − 1) = 0, the zero-product rule gives two cases. Dividing by sin x would discard every solution in the first case. Keep factors, solve each case, combine the solution sets, and remove duplicates only after checking the original interval.

When using sin² x + cos² x = 1, choose a substitution that reduces the equation to one circular function. Squaring an equation can introduce extra sign possibilities, so substitute candidates into the original unsquared equation. The graph can help estimate how many intersections to expect, but it cannot justify an exact angle or guarantee that a touching intersection was detected.

6. A periodic model needs a starting phase

For a sinusoidal model, the average of the maximum and minimum gives the midline, half their difference gives the amplitude, and the time between successive matching cycle positions gives the period. The starting observation determines the phase. A cosine model is often convenient when time zero is a maximum or minimum; a sine model can be convenient when it is a midline crossing with a known direction.

Keep units attached to the independent variable. If t is in hours, the period must also be in hours and the angle coefficient has the corresponding reciprocal-time scale. A formula can be periodic for all real time while the modelling question restricts it to a particular observation interval. Interpret “first”, “next” and “during the interval” before listing solutions.

Practise, then check

1. Convert 150 degrees to radians and 7π/4 radians to degrees.
2. Find exact values of sin(7π/6), cos(5π/3) and tan(3π/4).
3. State the amplitude, period, midline and range of y = −3 cos(4x) + 2.
4. Solve cos(2x) = 1/2 for 0 ≤ x ≤ π.
5. Find all vertical asymptotes of y = tan(2x − π/4), and state its period.
6. Solve sin x(2 cos x − 1) = 0 for 0 ≤ x < 2π.
7. A model has minimum 4, maximum 10, period 8 hours, and is at its maximum at t = 0. Give a cosine rule and find the first time it reaches its midline.

Exam habits and technology checks

Write the angle mode and the interval before solving. Use exact multiples of π when available, and check each endpoint separately. For a graph, label the period, midline, extrema and any asymptotes. For technology-assisted work, adjust the window to show the entire requested interval and use algebra to confirm a complete solution set rather than relying on one numerical intersection.