What does SOH CAH TOA stand for?

Sine = Opposite ÷ Hypotenuse, Cosine = Adjacent ÷ Hypotenuse, Tangent = Opposite ÷ Adjacent. It's a way to remember the three trigonometric ratios for right-angled triangles.

How do I know whether to use sin, cos or tan?

Label the sides from the angle you're using, then look at which two sides are involved (the one you know and the one you want). Opposite and hypotenuse: sin. Adjacent and hypotenuse: cos. Opposite and adjacent: tan.

Why is my calculator giving the wrong answer?

It's probably in radians mode. Australian school questions use degrees, so check the screen shows D or DEG.

When do I use Pythagoras instead of trigonometry?

If the question involves only sides (two known, one to find), use Pythagoras. If it involves an angle, either known or to find, use trigonometry.

What year level learns SOH CAH TOA?

In the Victorian Curriculum 2.0 it's introduced in Year 9 and applied in Year 10, including angles of elevation and depression and bearings. Year 10A adds the sine and cosine rules and 3D problems.

SOH CAH TOA Explained: Find Missing Sides and Angles, with Examples (Years 9 and 10)

Step 1: label the sides from the angle

Always label from the angle you're using, $\theta$ (not the right angle):

Hypotenuse (H): the longest side, opposite the right angle. It never changes.
Opposite (O): the side across from $\theta$ . It doesn't touch the angle.
Adjacent (A): the side next to $\theta$ that isn't the hypotenuse.

If you use the other angle in the triangle, the opposite and adjacent swap.

Step 2: pick the ratio

\sin\theta = \frac{\text{O}}{\text{H}}

\cos\theta = \frac{\text{A}}{\text{H}}

\tan\theta = \frac{\text{O}}{\text{A}}

You know or want	Use	Remember
Opposite and hypotenuse	$\sin$	SOH
Adjacent and hypotenuse	$\cos$	CAH
Opposite and adjacent	$\tan$	TOA

Look at the two sides in the question (the one you know and the one you want) and find the row that has both.

Trigonometry calculator (with working)

Find a side or an angle. You’ll get every step, set out the way your teacher wants to see it.

Angle θ (degrees)Side you knowIts lengthSide to findUnits (optional)

The opposite is about 6.88 cm

Label the sides from the angle $35^\circ$ : you know the hypotenuse and want the opposite.
The opposite and the hypotenuse go together in SOH: $\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$
Substitute: $\sin 35^\circ = \dfrac{x}{12}$
Multiply both sides by $12$ : $x = 12 \times \sin 35^\circ$
Use a calculator (in degrees mode): $\sin 35^\circ \approx 0.5736$
$x \approx 6.88 \text{ cm}$ (rounded to 2 decimal places)

Sides are labelled from the angle θ: the opposite is across from it, the adjacent is next to it, and the hypotenuse is the longest side, opposite the right angle.

Step 3: finding a missing side

If the unknown is on top of the fraction, multiply. If it’s on the bottom, divide.

Unknown on top

The angle is $35^\circ$ and the hypotenuse is 12 cm. Find the opposite side.

Opposite and hypotenuse, so SOH:

\sin 35^\circ = \frac{x}{12}

x = 12 \times \sin 35^\circ \approx 6.88 \text{ cm}

Unknown on the bottom

The angle is $40^\circ$ and the opposite side is 9 m. Find the adjacent side.

Opposite and adjacent, so TOA:

\tan 40^\circ = \frac{9}{x}

x = \frac{9}{\tan 40^\circ} \approx 10.73 \text{ m}

Finding the hypotenuse

The angle is $62^\circ$ and the adjacent side is 5 cm. Find the hypotenuse.

Adjacent and hypotenuse, so CAH:

\cos 62^\circ = \frac{5}{x}

x = \frac{5}{\cos 62^\circ} \approx 10.65 \text{ cm}

Finding a missing angle

With two sides, write the ratio as a fraction, then use the inverse key (shift, then sin, cos or tan).

Worked example

The opposite side is 7 and the adjacent side is 10. Find the angle.

\tan\theta = \frac{7}{10}

\theta = \tan^{-1}(0.7) \approx 35.0^\circ

Worked example

The opposite side is 4 and the hypotenuse is 9. Find the angle.

\sin\theta = \frac{4}{9}

\theta = \sin^{-1}\left(\frac{4}{9}\right) \approx 26.4^\circ

Angles of elevation and depression (Year 10)

The angle of elevation is measured up from the horizontal; the angle of depression is measured down from the horizontal. Draw the horizontal line first, then the right-angled triangle.

Worked example

From 30 m away, the angle of elevation to the top of a tree is $28^\circ$ . How tall is the tree?

The height is opposite the angle and the 30 m is adjacent, so TOA:

\tan 28^\circ = \frac{h}{30}

h = 30 \tan 28^\circ \approx 15.95 \text{ m}

Common mistakes that cost marks

Calculator in radians. Check for D or DEG on the screen before you start.
Labelling from the right angle. Opposite and adjacent depend on the angle θ you’re using.
Multiplying when the unknown is on the bottom. If x is the denominator, divide the known side by the ratio.
Forgetting the inverse to find an angle. sin θ = 0.44 means θ = sin⁻¹(0.44), not 0.44°.
Rounding the ratio too early. Keep the full calculator value, then round the final answer.

Practice questions with answers

Try each one before you open the answer. Calculator in degrees mode.

The angle is $50^\circ$ and the hypotenuse is 20 cm. Find the opposite side, to 2 decimal places.
Show the answer
$\sin 50^\circ = \dfrac{x}{20}$ , so $x = 20 \sin 50^\circ \approx 15.32$ cm.
The angle is $25^\circ$ and the adjacent side is 14 m. Find the opposite side, to 2 decimal places.
Show the answer
$\tan 25^\circ = \dfrac{x}{14}$ , so $x = 14 \tan 25^\circ \approx 6.53$ m.
The angle is $70^\circ$ and the opposite side is 8 cm. Find the hypotenuse, to 2 decimal places.
Show the answer
$\sin 70^\circ = \dfrac{8}{x}$ , so $x = \dfrac{8}{\sin 70^\circ} \approx 8.51$ cm.
The opposite side is 5 and the hypotenuse is 13. Find the angle, to 1 decimal place.
Show the answer
$\sin\theta = \dfrac{5}{13}$ , so $\theta = \sin^{-1}\left(\dfrac{5}{13}\right) \approx 22.6^\circ$ .
The adjacent side is 12 and the hypotenuse is 15. Find the angle, to 1 decimal place.
Show the answer
$\cos\theta = \dfrac{12}{15} = 0.8$ , so $\theta = \cos^{-1}(0.8) \approx 36.9^\circ$ .
A ramp rises 1.2 m over a horizontal distance of 6 m. What angle does it make with the ground, to 1 decimal place?
Show the answer
$\tan\theta = \dfrac{1.2}{6} = 0.2$ , so $\theta = \tan^{-1}(0.2) \approx 11.3^\circ$ .

Want more? Each year level below has a free printable PDF of 10 questions with answers, and a full 20-question test. Or check your working with the calculator.

Trigonometry by year level

Following the Victorian Curriculum 2.0. Schools sequence topics differently, so check your course.

Year 9

Trigonometry

Introducing sin, cos and tan: labelling sides and finding missing sides and angles.

Worksheets and topic test · Free PDF with answers

Year 10

Trigonometry

Applying trigonometry: angles of elevation and depression, bearings and multi-step problems.

Worksheets and topic test · Free PDF with answers

Year 10A

Sine rule, cosine rule and area

Beyond right angles: the sine rule, the cosine rule and the area of a triangle.

Worksheets and topic test · Free PDF with answers

Year 10A

Pythagoras and trigonometry in 3D

Pythagoras and trigonometry in 3D shapes.

Worksheets and topic test · Free PDF with answers

SOH CAH TOA: how to find missing sides and angles

Step 1: label the sides from the angle

Step 2: pick the ratio

Trigonometry calculator (with working)

Step 3: finding a missing side

Finding a missing angle

Angles of elevation and depression (Year 10)

Common mistakes that cost marks

Practice questions with answers

Trigonometry by year level

SOH CAH TOA: common questions

More maths help, Years 7 to 10

Got a trig question wrong in homework?