IMAT Mathematics · Free notes

Trigonometry: Ratios, Graphs and Equations

Trigonometry has 4 past questions since 2011, and the ministry papers from 2023 on have asked 3 of them. Each one rests on a small set of facts: which side goes with sine and which with cosine, the exact values at the standard angles, the signs in each quadrant, and the fact that sine and cosine never leave the interval from minus 1 to 1. The harder items are equations, where you must count every solution in a given range, and expressions that turn into a quadratic in sin x. No calculator is allowed, so the exact values must be automatic.

The handout has every subtopic, worked example, formula and trap with answers shown. The formula sheet is the chapter’s formulas and traps alone, on one page.

IMAT home All formulas on one page

Subtopic notes

Formula & revision sheet

6 formulas · 1 reference tables · 8 gotchas across all subtopics — the exam-eve cheat-sheet

Angles, Ratios and Exact Values

Formulas (3)

Measuring angles in radians and degrees · Converting angles $\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180} \qquad s = r\theta$
Sine, cosine and tangent in a right triangle · SOH CAH TOA $\sin\alpha = \frac{\text{opp}}{\text{hyp}} \qquad \cos\alpha = \frac{\text{adj}}{\text{hyp}} \qquad \tan\alpha = \frac{\text{opp}}{\text{adj}}$
The unit circle: any angle, signs in four quadrants and sin² + cos² = 1 · The two basic identities $\sin^2\theta + \cos^2\theta = 1 \qquad \tan\theta = \frac{\sin\theta}{\cos\theta}$

Reference tables (1)

Exact values of sine, cosine and tangent at the standard angles5 rows

Angle	sin	cos	tan
$0^\circ$ (0)	0	1	0
$30^\circ$ $(\pi/6)$	$\dfrac{1}{2}$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{\sqrt{3}}{3}$
$45^\circ$ $(\pi/4)$	$\dfrac{\sqrt{2}}{2}$	$\dfrac{\sqrt{2}}{2}$	1
$60^\circ$ $(\pi/3)$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{2}$	$\sqrt{3}$
$90^\circ$ $(\pi/2)$	1	0	Not defined $\tan 90^\circ$ does not exist, because $\cos 90^\circ = 0$ .

From the triangles with sides

1, \sqrt{3}, 2

and

1, 1, \sqrt{2}

Watch out for (4)

Pi radians is 180 degrees, not 360
→ Measuring angles in radians and degrees
Opposite goes with sine, adjacent with cosine
→ Sine, cosine and tangent in a right triangle
sin 30° is one half; cos 30° is root 3 over 2
→ Exact values of sine, cosine and tangent at the standard angles
The identity gives cos only up to a sign
→ The unit circle: any angle, signs in four quadrants and sin² + cos² = 1

Graphs, Equations and the Sine and Cosine Rules

Formulas (3)

Graphs of sine, cosine and tangent: period and amplitude · Period and amplitude $y = a\sin(bx) + c: \quad \text{amplitude } |a|, \quad \text{period } \frac{360^\circ}{b}$
Solving trigonometric equations in a given range · General solutions $\sin x = k: \ x = \alpha,\ 180^\circ - \alpha \qquad \cos x = k: \ x = \pm\alpha \qquad \tan x = k: \ x = \alpha \quad (+\text{whole periods})$
The sine rule and the cosine rule for any triangle · Sine rule, cosine rule and area $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \qquad c^2 = a^2 + b^2 - 2ab\cos C \qquad \text{Area} = \tfrac{1}{2}ab\sin C$

Watch out for (4)

The period of sin(bx) is 360° divided by b
→ Graphs of sine, cosine and tangent: period and amplitude
Solve for the whole angle first
→ Solving trigonometric equations in a given range
Throw away values of sin x outside minus 1 to 1
→ Solving trigonometric equations in a given range
The cosine rule uses the angle between the two sides
→ The sine rule and the cosine rule for any triangle

PYQ weightage by concept

7 concepts · 0 PYQs — where the marks actually sit, so you know what to drill first

Angles, Ratios and Exact Values0 PYQs · 0%

Concept	PYQs	Share
Measuring angles in radians and degreesfoundation	—	—
Sine, cosine and tangent in a right trianglefoundation	—	—
Exact values of sine, cosine and tangent at the standard anglesfoundation	—	—
The unit circle: any angle, signs in four quadrants and sin² + cos² = 1foundation	—	—

Graphs, Equations and the Sine and Cosine Rules0 PYQs · 0%

Concept	PYQs	Share
Graphs of sine, cosine and tangent: period and amplitudefoundation	—	—
Solving trigonometric equations in a given rangefoundation	—	—
The sine rule and the cosine rule for any trianglefoundation	—	—