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Right-Angled Trigonometry (OCR GCSE Maths)
Revision Note
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 hypotenuse, H, is the longest side in a right-angled triangle
- The functions Sine, Cosine and Tangent are the ratios of the lengths of these sides as follows
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
- 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:
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:
, ,
- After choosing the correct ratio and substituting the values use the inverse trigonometric functions on your calculator to find the correct answer
Examiner 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 and in the following triangles.
Give your answers to 3 significant figures.
To find , first label the triangle
We know A and we want to know O - that's TOA or
Multiply both sides by 9
Enter on your calculator
Round to 3 significant figures
To find , first label the triangle
We know A and H - that's CAH or
Use inverse cos to find
Enter on your calculator
Round to 3 significant figures
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
Examiner 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 m up the cliff is a flag marker and the angle of elevation from the boat to the flag marker is 18°.
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