Source: ACE Exam Paper 1 ADV
Tags: trigonometry, trig equations, cosine, tangent, amplitude, period, phase shift, trig graphs, unit circle, absolute value trig
The exam tests trig equations (solving in a domain), reading and writing trig function equations from graphs, and properties like amplitude and period. Most questions hinge on knowing exact values, understanding transformations (stretch, shift), and being methodical about finding all solutions in the given domain.
Amplitude
The distance from the centre line to the peak (or trough) of a trig function. For y = A + B cos(Cx), the amplitude is |B|.
Period
The horizontal length of one complete cycle. For y = cos(Cx), the period is 2π/|C|. For y = tan(Cx), the period is π/|C|.
Phase shift
A horizontal translation of a trig graph. In y = sin(C(x - D)), the graph shifts right by D units.
Asymptote (for tangent)
Vertical lines where tan is undefined. For y = tan(C(x - D)), asymptotes occur where C(x - D) = π/2 + nπ for integer n.
Rearrange: cos²x = 1/2, so cos x = ±1/√2 = ±√2/2.
cos x = √2/2: x = π/4, x = 7π/4
cos x = -√2/2: x = 3π/4, x = 5π/4
All four solutions: x = π/4, 3π/4, 5π/4, 7π/4. The answer is (D).
Note: option (C) lists 11π/4, which exceeds 2π, so it is outside the domain and can be eliminated immediately.
For a tan graph, you need to read two things: the period and the phase shift.
If the graph has vertical asymptotes at x = -π/2 and x = π/2 (among others), that is a period of π, which means the coefficient of x inside the tan is 1. If asymptotes are at x = 0 and x = π, the period is still π but there is a phase shift.
For the exam question: the graph shows a tan function with a period of 2π (asymptotes spaced π apart when the coefficient is halved). This means the coefficient is 1/2, giving tan((1/2)(...)). The asymptotes and the position of the zero crossing tell you the phase shift.
If the graph passes through the origin shifted to x = π/4 with asymptotes suggesting a half-frequency tangent shifted by π/4, the function is f(x) = tan((1/2)(x - π/4)). Answer (A).
If the period appears to be π/2 instead, the coefficient is 2, and you pick the corresponding option. Read the asymptote spacing carefully.
Amplitude = |3| = 3
The "4" shifts the graph vertically but does not affect the amplitude.
Period = 2π / (π/2) = 2π × (2/π) = 4
|cos(2x)| = 1 means cos(2x) = 1 or cos(2x) = -1.
Let u = 2x, so 0 ≤ u ≤ 4π.
cos u = 1 at u = 0, 2π, 4π (three solutions)
cos u = -1 at u = π, 3π (two solutions)
Total: 5 solutions.
The corresponding x-values are x = 0, π/2, π, 3π/2, 2π.
cos²x = (1 + cos 2x)/2
sin²x = (1 - cos 2x)/2
Period of y = A sin(Bx + C) or A cos(Bx + C): 2π/|B|
Period of y = A tan(Bx + C): π/|B|
Amplitude of y = A sin(Bx + C): |A|
Exact values: cos(π/4) = sin(π/4) = √2/2, cos(π/3) = 1/2, sin(π/3) = √3/2
⚠️ Always check that all your solutions fall within the stated domain. Solutions outside the domain score zero marks and signal carelessness.
⚠️ When solving cos²x = k, take both the positive and negative square root. Forgetting one halves your solution set.
⚠️ For tangent graph identification, read the asymptote positions first. The period is the distance between consecutive asymptotes. The zero crossing halfway between asymptotes confirms the phase shift.
⚠️ Vertical shifts (like the +4 in 4 + 3cos(...)) do not change the amplitude or period. They only move the centre line.
⚠️ |cos(2x)| = 1 is different from cos(2x) = 1. The absolute value doubles the solution count from cos = 1 alone.
Q: Solve 2cos²x - 1 = 0 for 0 ≤ x ≤ 2π.
A: cos x = ±√2/2, giving x = π/4, 3π/4, 5π/4, 7π/4.
Q: State the amplitude and period of f(x) = 4 + 3cos(πx/2).
A: Amplitude = 3. Period = 4.
Q: How many solutions does |cos(2x)| = 1 have for 0 ≤ x ≤ 2π?
A: Five. At x = 0, π/2, π, 3π/2, 2π.
Q: A tangent graph has consecutive asymptotes at x = π/4 and x = 5π/4. What is the period?
A: Period = 5π/4 - π/4 = π. The coefficient of x inside the tangent is π/π = 1.
trigonometric equations, trig graphs, cosine equation, tangent graph, amplitude, period, phase shift, frequency, unit circle, exact values, vertical asymptote, trig transformations, absolute value cosine, HSC maths advanced, Year 12 trigonometry