Right-Angled Trigonometry

SOHCAHTOA

What is Trigonometry?

Trigonometry is the mathematics of angles in triangles
It looks at the relationship between side lengths and angles of triangles
It comes from the Greek words trigonon meaning ‘triangle’ and metron meaning ‘measure’

What are Sin, Cos and Tan?

The three trigonometric functions Sine, Cosine and Tangent come from ratios of side lengths in right-angled triangles
To see how the ratios work you must first label the sides of a right-angled triangle in relation to a chosen angle
- The hypotenuse, H, is the longest side in a right-angled triangle
  - It will always be opposite the right angle
- If we label one of the other angles θ, the side opposite θ will be labelled opposite, O, and the side next to θ will be labelled adjacent, A
The functions Sine, Cosine and Tangent are the ratios of the lengths of these sides as follows

$Sin θ = \frac{opposite}{hypotenuse} = \frac{O}{H}$

$Cos θ = \frac{adjacent}{hypotenuse} = \frac{A}{H}$

$Tan θ = \frac{opposite}{adjacent} = \frac{O}{A}$

What is SOHCAHTOA?

SOHCAHTOA is a mnemonic that is often used as a way of remembering which ratio is which
- Sin is Opposite over Hypotenuse
- Cos is Adjacent over Hypotenuse
- Tan is Opposite over Adjacent
In a right-angled triangle, label one angle other than the right angle and label the sides of the triangles as follows

Right-Angled-Triangles-OAH-Theta, IGCSE & GCSE Maths revision notes

Note that θ is the Greek letter theta
- O = opposite θ
- A = adjacent (next to) θ
- H = hypotenuse - 'H' is always the same, but 'O' and 'A' change depending on which angle we're calling θ
Using those labels, the three SOHCAHTOA equations are:

Right-Angled Triangles Diagram 1

How can we use SOHCAHTOA to find missing lengths?

If you know the length of one of the sides of any right-angled triangle and one of the angles you can use SOHCAHTOA to find the length of the other sides
- Always start by labelling the sides of the triangle with H, O and A
- Choose the correct ratio by looking only at the values that you have and that you want
  - For example if you know the angle and the side opposite it (O) and you want to find the hypotenuse (H) you should use the sine ratio
- Substitute the values into the ratio
- Use your calculator to find the solution

How can we use SOHCAHTOA to find missing angles?

If you know two sides of any right-angled triangle you can use SOHCAHTOA to find the size of one of the angles
Missing angles are found using the inverse functions:

$θ = {Sin}^{- 1} \frac{O}{H}$ , $θ = {Cos}^{- 1} \frac{A}{H}$ , $θ = {Tan}^{- 1} \frac{O}{A}$

After choosing the correct ratio and substituting the values use the inverse trigonometric functions on your calculator to find the correct answer

Exam Tip

SOHCAHTOA (like Pythagoras) can only be used in right-angles triangles – for triangles that are not right-angled, you will need to use the Sine Rule or the Cosine Rule
Also, make sure your calculator is set to measure angles in degrees

Worked example

Find the values of $x$ and $y$ in the following triangles.

Give your answers to 3 significant figures.

To find $x$ , first label the triangle

We know A and we want to know O - that's TOA or $tanθ = \frac{opposite}{adjacent}$

$\tan (43) = \frac{x}{9}$

Multiply both sides by 9

$9 \times \tan (43) = x$

Enter on your calculator

$x = 8.3926 . . .$

Round to 3 significant figures

$x = 8.39 cm$

To find $y$ , first label the triangle

We know A and H - that's CAH or $cosθ = \frac{adjacent}{hypotenuse}$

$\cos (y) = \frac{8}{23}$

Use inverse cos to find $y$

$y = \cos^{- 1} (\frac{8}{23})$

Enter on your calculator

$y = 69.6455 . . .$

Round to 3 significant figures

$y = 69.6 °$

Elevation & Depression

What are the angles of elevation and depression?

If a person looks at an object that is not on the same horizontal line as their eye-level they will be looking at either an angle of elevation or depression
- If a person looks up at an object their line of sight will be at an angle of elevation with the horizontal
- If a person looks down at an object their line of sight will be at an angle of depression with the horizontal
Angles of elevation and depression are measured from the horizontal
Right-angled trigonometry can be used to find an angle of elevation or depression or a missing distance
Tan is often used in real-life scenarios with angles of elevation and depression
- For example if we know the distance we are standing from a tree and the angle of elevation of the top of the tree we can use Tan to find its height
- Or if we are looking at a boat at to sea and we know our height above sea level and the angle of depression we can find how far away the boat is

Exam Tip

It may be useful to draw more than one diagram if the triangles that you are interested in overlap one another

Worked example

A cliff is perpendicular to the sea and the top of the cliff stands 24 m above the level of the sea. The angle of depression from the cliff to a boat at sea is 35°. At a point $x$ m up the cliff is a flag marker and the angle of elevation from the boat to the flag marker is 18°.

Draw and label a diagram to show the top of the cliff, T, the foot of the cliff, F, the flag marker, M, and the boat, B, labelling all the angles and distances given above.

Find the distance from the boat to the foot of the cliff.

Find the value of

x

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