Right-Angled Triangles

Lessons

Senior 3 · Mathematics

Right-Angled Triangles

5 lessons

Lesson 3 of 5

Sine and cosine of an acute angle

25 min

Same angle, same ratio

Any two right-angled triangles with the same acute angle are similar: one is an enlargement of the other. So the ratio of any two of their sides is the same, however big the triangle. For an angle of $30^\circ$ , the side opposite the angle is always exactly half the hypotenuse, whether the triangle is drawn in an exercise book or marked out on a building site.

Doubling the triangle doubles every side, so the ratio of two sides does not change.

This lesson gives names to these fixed ratios, the sine and the cosine, and uses them to find lengths and angles that cannot be measured directly.

Definition

Opposite, adjacent and hypotenuse: in a right-angled triangle, fix one acute angle $\theta$ (theta). The hypotenuse is opposite the right angle; the opposite side is across from $\theta$ ; the adjacent side is the leg next to $\theta$ .

The names depend on which angle you choose: for the angle at B, the opposite and adjacent sides swap.

Definition

Sine and cosine of an acute angle $\theta$ in a right-angled triangle:

\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \qquad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}.

Notation: $\sin\theta$ and $\cos\theta$ are numbers without units. Because the hypotenuse is the longest side, $0 < \sin\theta < 1$ and $0 < \cos\theta < 1$ for every acute angle.

Example: in the 3, 4, 5 triangle, for the angle opposite the side 3, $\sin\theta = \frac{3}{5} = 0.6$ and $\cos\theta = \frac{4}{5} = 0.8$ .

Important

Exact values for $30^\circ$ , $45^\circ$ and $60^\circ$ . Cut an equilateral triangle of side 2 in half: the half has hypotenuse 2, short side 1 and, by Pythagoras, height $\sqrt{2^2 - 1^2} = \sqrt{3}$ . An isosceles right-angled triangle with legs 1 has hypotenuse $\sqrt{2}$ .

Half an equilateral triangle, and half a square.

$\theta$	$30^\circ$	$45^\circ$	$60^\circ$
$\sin\theta$	$\frac{1}{2}$	$\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$
$\cos\theta$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$

Notice that $\sin 30^\circ = \cos 60^\circ$ . In general $\sin\theta = \cos(90^\circ - \theta)$ , because the side opposite $\theta$ is adjacent to the other acute angle, $90^\circ - \theta$ .

Sine and cosine on a calculator

Check that the calculator shows D or DEG (degrees). Then press sin 35 = to get $\sin 35^\circ \approx 0.5736$ . To go back from a ratio to an angle, use the inverse key, SHIFT sin, written $\sin^{-1}$ : $\sin^{-1}(0.6) \approx 36.87^\circ$ .

$\theta$	$10^\circ$	$20^\circ$	$30^\circ$	$40^\circ$	$50^\circ$	$60^\circ$	$70^\circ$	$80^\circ$
$\sin\theta$	0.1736	0.3420	0.5000	0.6428	0.7660	0.8660	0.9397	0.9848
$\cos\theta$	0.9848	0.9397	0.8660	0.7660	0.6428	0.5000	0.3420	0.1736

As $\theta$ grows from $0^\circ$ to $90^\circ$ , the sine grows and the cosine shrinks, and the two rows are mirror images: $\sin 20^\circ = \cos 70^\circ$ .

Worked example 1: finding two sides

Find $x$ and $y$ .

Solution $x$ is opposite $35^\circ$ and 12 cm is the hypotenuse, so use sine: $\sin 35^\circ = \frac{x}{12}$ , giving $x = 12\sin 35^\circ \approx 6.88$ cm.

$y$ is adjacent, so use cosine: $y = 12\cos 35^\circ \approx 9.83$ cm.

Check with Pythagoras: $6.883^2 + 9.830^2 \approx 47.4 + 96.6 = 144 = 12^2$ ✓.

Worked example 2: finding an angle

In a right-angled triangle, the side opposite $\theta$ is 5 cm and the hypotenuse is 8 cm. Find $\theta$ .

Solution $\sin\theta = \frac{5}{8} = 0.625$ , so $\theta = \sin^{-1}(0.625) \approx 38.7^\circ$ (to 1 decimal place).

Worked example 3: a ramp for a wheelchair

A school builds a ramp to a doorway 0.6 m above the ground. The ramp must make an angle of $5^\circ$ with the ground (an illustrative design limit). How long is the sloping surface of the ramp?

Drawn to scale: a 5° ramp is long and gentle.

Solution The rise is opposite $5^\circ$ and the ramp is the hypotenuse $L$ : $\sin 5^\circ = \frac{0.6}{L}$ , so

L = \frac{0.6}{\sin 5^\circ} \approx \frac{0.6}{0.0872} \approx 6.88\ \text{m}.

When the unknown is on the bottom of the fraction, rearrange before you use the calculator: $L = \frac{\text{opposite}}{\sin\theta}$ .

Warning

Degrees, not radians. If $\sin 30$ gives $- 0.988$ , your calculator is in radians (R or RAD). Switch it to D before you start.

$\sin^{-1}$ is not $\frac{1}{\sin}$ . $\sin^{-1}(0.625)$ is the angle whose sine is 0.625.

Name the sides from the angle you are using. In Worked example 1, $y$ is adjacent to $35^\circ$ but opposite the other acute angle, $55^\circ$ .

Tip

SOH, CAH: Sine is Opposite over Hypotenuse; Cosine is Adjacent over Hypotenuse. Label the three sides O, A and H on the diagram before choosing the ratio.

Khan Academy: Basic trigonometry. Watch how the sides are named opposite, adjacent and hypotenuse relative to an angle, and how SOH CAH TOA defines the ratios.

✦

Practice

Give lengths to 3 significant figures and angles to 1 decimal place. The last three are harder.

A right-angled triangle has sides 5, 12 and 13. For the angle $\theta$ opposite the side 5, write $\sin\theta$ and $\cos\theta$ as fractions.
Find the exact value of $\sin 60^\circ \times \cos 30^\circ$ .
Use a calculator to find $\sin 52^\circ$ and $\cos 52^\circ$ to 4 decimal places.
The hypotenuse is 20 cm and one angle is $25^\circ$ . Find the side opposite that angle.
The side adjacent to $\theta$ is 9 cm and the hypotenuse is 15 cm. Find $\theta$ .
The side opposite a $38^\circ$ angle is 4.2 m. Find the hypotenuse.
A ladder 5 m long leans against a wall at $70^\circ$ to the ground. How high up the wall does it reach, and how far is its foot from the wall?
Without a calculator, explain why $\sin 25^\circ = \cos 65^\circ$ .

Show answersHide answers

$\sin\theta = \frac{5}{13}$ , $\cos\theta = \frac{12}{13}$ .
$\frac{\sqrt{3}}{2} \times \frac{\sqrt{3}}{2} = \frac{3}{4}$ .
$\sin 52^\circ \approx 0.7880$ , $\cos 52^\circ \approx 0.6157$ .
$20\sin 25^\circ \approx 8.45$ cm.
$\cos\theta = 0.6$ , so $\theta \approx 53.1^\circ$ .
$\frac{4.2}{\sin 38^\circ} \approx 6.82$ m.
Height $5\sin 70^\circ \approx 4.70$ m; foot $5\cos 70^\circ \approx 1.71$ m from the wall.
In a right-angled triangle with a $25^\circ$ angle, the other acute angle is $65^\circ$ . The side opposite $25^\circ$ is adjacent to $65^\circ$ , so both ratios equal that side divided by the hypotenuse.

Sources and credits

Written by Ganzaa.
Video: Basic trigonometry, Khan Academy, embedded from YouTube.

The diagrams, tables, worked examples, practice items and quiz questions were written by Ganzaa. The ramp angle is illustrative, not a building regulation.

1Same angle, same ratio

2Sine and cosine on a calculator

Same angle, same ratio

Sine and cosine on a calculator