Open Math›Year 9›Trigonometry›Sin, cos and tan ratios

Sin, cos and tan ratios

Identify the opposite, adjacent and hypotenuse sides of a right-angled triangle relative to a given angle, and use the sin, cos and tan ratios.

Worked examples

Identifying sides and writing ratios

Straightforward

Problem

A right-angled triangle has angle

\theta

at one vertex. The side opposite

\theta

6

cm, the side adjacent to

\theta

8

cm, and the hypotenuse is

10

cm. Write the values of

\sin \theta

\cos \theta

and

\tan \theta

Identify the three sides relative to

\theta

\text{opposite} = 6 \text{ cm}, \quad \text{adjacent} = 8 \text{ cm}, \quad \text{hypotenuse} = 10 \text{ cm}

Apply SOHCAHTOA to write

\sin \theta

\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{6}{10} = 0.6

Apply SOHCAHTOA to write

\cos \theta

\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{8}{10} = 0.8

Apply SOHCAHTOA to write

\tan \theta

\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{6}{8} = 0.75

Answer

\sin \theta = 0.6

\cos \theta = 0.8

\tan \theta = 0.75

Evaluating a trig ratio with a calculator

Moderate

Problem

Use a calculator to find

\tan 57°

correct to 4 decimal places.

Check that your calculator is set to degree mode.

Mode: Degrees

Enter

\tan 57°

on your calculator.

\tan 57° = 1.539864957\ldots

Round to 4 decimal places. The fifth decimal digit is

6

, so round up.

\tan 57° \approx 1.5399

Answer

\tan 57° \approx 1.5399

Setting up a trig equation for an unknown side

Challenging

Problem

A right-angled triangle has angle

\theta = 42°

at one vertex and a hypotenuse of

15

cm. Write an equation for the length of the opposite side

x

, then calculate

x

correct to 2 decimal places.

Identify the known information and the unknown. We know the hypotenuse and the angle, and we want the opposite side.

\text{hypotenuse} = 15 \text{ cm}, \quad \theta = 42°, \quad \text{opposite} = x

Choose the ratio that links the opposite side and the hypotenuse. That is SOH:

\sin \theta = \dfrac{\text{opposite}}{\text{hypotenuse}}

\sin 42° = \frac{x}{15}

Rearrange to make

x

the subject.

x = 15 \sin 42°

Evaluate using a calculator.

x = 15 \times 0.6691\ldots = 10.037\ldots \approx 10.04 \text{ cm}

Answer

x = 15 \sin 42° \approx 10.04

Practise

Q1·Straightforward

In a right-angled triangle, which side is the hypotenuse?

The side opposite the right angleThe side opposite angle $\theta$The side next to angle $\theta$ (not the hypotenuse)The shortest side

Explanation

The hypotenuse is the side directly across from the right angle (

90°

). It is always the longest side of the right-angled triangle.

Q2·Straightforward

In a right-angled triangle with angle

\theta

at one vertex, the side directly across from

\theta

is called the:

hypotenuseadjacent sideopposite sidebase

Explanation

The opposite side is the side directly across from the angle

\theta

. The adjacent side is the other non-hypotenuse side, which is next to

\theta

Q3·Straightforward

Which trigonometric ratio is defined as

\dfrac{\text{opposite}}{\text{hypotenuse}}

$\sin \theta$$\cos \theta$$\tan \theta$$\dfrac{1}{\cos \theta}$

Explanation

SOH stands for Sine = Opposite over Hypotenuse, so

\sin \theta = \dfrac{\text{opposite}}{\text{hypotenuse}}

Q4·Straightforward

Which trigonometric ratio is defined as

\dfrac{\text{opposite}}{\text{adjacent}}

$\sin \theta$$\cos \theta$$\tan \theta$$\dfrac{1}{\sin \theta}$

Explanation

TOA stands for Tangent = Opposite over Adjacent, so

\tan \theta = \dfrac{\text{opposite}}{\text{adjacent}}

Q5·Moderate

Use a calculator to find

\sin 35°

correct to 4 decimal places.

Explanation

Entering

\sin 35°

on a calculator gives

0.573576\ldots

, which rounds to

0.5736

to 4 decimal places.

Q6·Moderate

Use a calculator to find

\cos 48°

correct to 4 decimal places.

Explanation

Entering

\cos 48°

on a calculator gives

0.669131\ldots

, which rounds to

0.6691

to 4 decimal places.

Q7·Moderate

Use a calculator to find

\tan 62°

correct to 4 decimal places.

Explanation

Entering

\tan 62°

on a calculator gives

1.880726\ldots

, which rounds to

1.8807

to 4 decimal places.

Q8·Moderate

Use a calculator to find

\sin 72°

correct to 4 decimal places.

Explanation

Entering

\sin 72°

on a calculator gives

0.951056\ldots

, which rounds to

0.9511

to 4 decimal places.

Q9·Challenging

A right-angled triangle has angle

\theta = 40°

at one vertex, and the hypotenuse is

10

cm. Which equation correctly gives the length of the side opposite

\theta

$x = 10 \sin 40°$$x = 10 \cos 40°$$x = 10 \tan 40°$$x = \dfrac{10}{\sin 40°}$

Explanation

Since

\sin \theta = \dfrac{\text{opposite}}{\text{hypotenuse}}

, we have

\sin 40° = \dfrac{x}{10}

, so

x = 10 \sin 40°

Q10·Challenging

A right-angled triangle has angle

\theta = 55°

and the adjacent side is

8

cm. Which equation correctly gives the length of the hypotenuse

h

$h = \dfrac{8}{\cos 55°}$$h = 8 \cos 55°$$h = 8 \sin 55°$$h = \dfrac{8}{\sin 55°}$

Explanation

Using CAH:

\cos 55° = \dfrac{8}{h}

. Rearranging gives

h = \dfrac{8}{\cos 55°}

Q11·Challenging

In a right-angled triangle, the side opposite angle

\theta

7

cm and the adjacent side is

5

cm. Which equation correctly gives

\theta

$\theta = \tan^{-1}\!\left(\dfrac{7}{5}\right)$$\theta = \sin^{-1}\!\left(\dfrac{7}{5}\right)$$\theta = \cos^{-1}\!\left(\dfrac{5}{7}\right)$$\theta = \tan^{-1}\!\left(\dfrac{5}{7}\right)$

Explanation

Since

\tan \theta = \dfrac{\text{opposite}}{\text{adjacent}} = \dfrac{7}{5}

, we solve for

\theta

using the inverse:

\theta = \tan^{-1}\!\left(\dfrac{7}{5}\right)

Q12·Challenging

A right-angled triangle has angle

\theta = 30°

and hypotenuse

12

cm. Which expression gives the length of the adjacent side?

$12 \cos 30°$$12 \sin 30°$$\dfrac{12}{\cos 30°}$$\dfrac{\cos 30°}{12}$

Explanation

Using CAH:

\cos 30° = \dfrac{\text{adjacent}}{12}

. Multiplying both sides by

12

gives adjacent

= 12 \cos 30°