SOHCAHTOA - Finding Lengths
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 θ
- The hypotenuse, H, is the longest side in a right-angled triangle
- It will always be opposite the right angle
- The side opposite θ will be labelled opposite, O
- 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
- H is always the same but O and A change depending on which angle is labelled as θ
How can I use SOHCAHTOA to find missing lengths?
- STEP 1
Label the sides of the triangle as H, O and A- H is the longest side opposite the right angle
- O is opposite the given angle
- A is next to the given angle
- STEP 2
Identify which trigonometric ratio to use: sin, cos or tan- Write down the letter of the length you are given
- Write down the letter of the length you want to find
- Find the two letters in SOHCAHTOA to identify which ratio to use
- If you have A and H then use cos
- STEP 3
Substitute the values into the relevant trigonometric formula- Remember to put brackets around the angle
- or
- Remember to put brackets around the angle
- STEP 4
Rearrange and solve for the unknown letter- You will either need to multiply or divide
- leads to
- leads to
- You will either need to multiply or divide
- STEP 5
Type the expression into your calculator- The question might ask you to round your answer
- If not then round to three significant figures
Examiner Tip
- SOHCAHTOA (like Pythagoras) can only be used in right-angles triangles
- Ensure your calculator is set to measure angles in degrees
- You should see the letter D or the word Deg at the top of your screen
Worked example
Find the length of the side cm in the following triangle.
Give your answer to 3 significant figures.
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