IMAT Mathematics · Trigonometry

Angles, Ratios and Exact Values

Sine, cosine and tangent compare the sides of a right triangle, extend to any angle on the unit circle, and have exact values at 0°, 30°, 45°, 60° and 90°.

All Trigonometry notes

Why this matters

The 2024 paper asked which side of a right triangle equals the hypotenuse times the sine of an angle, and a 2021 question combined a tangent and a cosine across two right triangles sharing a side.

Concept 1 of 4: Measuring angles in radians and degrees

Wrap the radius of a circle along its edge: the angle it covers is one radian, a little under 60°. The whole edge is

2\pi

radii long, so a full turn is

2\pi

radians. Radians make arc lengths simple, which is why advanced formulas use them.

Definition

A full turn is $360^\circ = 2\pi$ radians, so $180^\circ = \pi$ radians.
Degrees to radians: multiply by $\dfrac{\pi}{180}$ . Radians to degrees: multiply by $\dfrac{180}{\pi}$ .
One radian is about $57.3^\circ$ .
With the angle $\theta$ in radians, an arc of a circle of radius $r$ has length $s = r\theta$ .

Converting angles

\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180} \qquad s = r\theta

$\theta$ the angle
$s$ arc length, with $\theta$ in radians
$r$ radius

Worked example

Convert

150^\circ

to radians and

\dfrac{3\pi}{4}

radians to degrees.

Practice this conceptself-check · 4 quick reps

Pi radians is 180 degrees, not 360

A full turn is

2\pi

, so

\pi

alone is half a turn,

180^\circ

. Using

360^\circ

for

\pi

doubles every converted angle.

Concept 2 of 4: Sine, cosine and tangent in a right triangle

All right triangles with the same acute angle are similar, so the ratios of their sides depend only on that angle. Sine, cosine and tangent are names for those ratios. Turned round, they give a side: the side opposite an angle is the hypotenuse times the sine.

Definition

In a right triangle, for an acute angle $\alpha$ :

The hypotenuse is the longest side, opposite the right angle; the opposite side faces $\alpha$ ; the adjacent side is the leg next to $\alpha$ .
$\sin\alpha = \dfrac{\text{opposite}}{\text{hypotenuse}}$ , $\cos\alpha = \dfrac{\text{adjacent}}{\text{hypotenuse}}$ , $\tan\alpha = \dfrac{\text{opposite}}{\text{adjacent}}$ (SOH CAH TOA).
So opposite $= \text{hyp} \times \sin\alpha$ , adjacent $= \text{hyp} \times \cos\alpha$ , opposite $= \text{adjacent} \times \tan\alpha$ .
The other acute angle is $90^\circ - \alpha$ , and $\sin\alpha = \cos(90^\circ - \alpha)$ .

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}}

Worked example

A right triangle has legs 8 cm and 15 cm. Let

\alpha

be the angle opposite the 8 cm leg. Find

\sin\alpha

\cos\alpha

and

\tan\alpha

Practice this conceptself-check · 3 quick reps

Opposite goes with sine, adjacent with cosine

The leg facing the angle is hypotenuse times sine; the leg touching the angle is hypotenuse times cosine. Swapping them is the most common wrong option, so label the triangle from the given angle before writing anything.

Concept 3 of 4: Exact values of sine, cosine and tangent at the standard angles

Two triangles give all these values. Half an equilateral triangle of side 2 has sides 1,

\sqrt{3}

and 2 with angles 30°, 60°, 90°. A right isosceles triangle with legs 1 has hypotenuse

\sqrt{2}

and angles 45°, 45°, 90°.

Definition

Sine rises from 0 to 1 as the angle goes from 0° to 90°; cosine falls from 1 to 0 over the same range.
$\sin\theta = \cos(90^\circ - \theta)$ , so the sine column read downwards is the cosine column read upwards.
$\dfrac{1}{\sqrt{3}} = \dfrac{\sqrt{3}}{3}$ and $\dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{2}}{2}$ .

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}

Practice this conceptself-check · 4 quick reps

sin 30° is one half; cos 30° is root 3 over 2

The two values swap between 30° and 60°. Remember that sine grows with the angle: the smaller angle has the smaller sine, so

\sin 30^\circ = \tfrac{1}{2}

and

\sin 60^\circ = \tfrac{\sqrt{3}}{2}

Concept 4 of 4: The unit circle: any angle, signs in four quadrants and sin² + cos² = 1

Put a point on a circle of radius 1 and turn it anticlockwise from the positive x-axis by

\theta

. Its x-coordinate is

\cos\theta

and its y-coordinate is

\sin\theta

. This works for any angle, and Pythagoras on the radius gives the most useful identity in trigonometry.

Definition

On the unit circle the point at angle $\theta$ is $(\cos\theta, \sin\theta)$ , so both lie between $-1$ and $1$ .
Signs: first quadrant (0° to 90°) all positive; second (90° to 180°) only sine positive; third (180° to 270°) only tangent positive; fourth (270° to 360°) only cosine positive.
The reference angle is the acute angle to the x-axis: $\sin 150^\circ = \sin 30^\circ$ , $\cos 240^\circ = -\cos 60^\circ$ .
Identities: $\sin^2\theta + \cos^2\theta = 1$ and $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$ .

The two basic identities

\sin^2\theta + \cos^2\theta = 1 \qquad \tan\theta = \frac{\sin\theta}{\cos\theta}

Worked example

The angle

\theta

is between

90^\circ

and

180^\circ

, and

\sin\theta = \dfrac{3}{5}

. Find

\cos\theta

and

\tan\theta

Practice this conceptself-check · 4 quick reps

The identity gives cos only up to a sign

\sin^2\theta + \cos^2\theta = 1

gives

\cos\theta = \pm\sqrt{1 - \sin^2\theta}

. The quadrant decides the sign, and the option with the wrong sign is always offered.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

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

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