Trigonometry to Find Lengths (Cambridge (CIE) IGCSE International Maths)

Revision Note

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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 functions sine, cosine and tangent are the ratios of the lengths of these sides as follows

    • sin space theta blank equals space opposite over hypotenuse space equals space straight O over straight H

    • cos space theta blank equals space adjacent over hypotenuse space equals space straight A over straight H

    • tan space theta blank equals space opposite over adjacent space equals space straight O over straight A

Trigonometric formulae for a right-angled triangle

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

Labelling a right-angled triangle using H, O and A

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

      • sin open parentheses 50 close parentheses equals straight A over 7  or  cos open parentheses 40 close parentheses equals 3 over straight H

  • STEP 4
    Rearrange and solve for the unknown letter

    • You will either need to multiply or divide

      • sin open parentheses 50 close parentheses equals straight A over 7 leads to straight A equals 7 cross times sin open parentheses 50 close parentheses

      • cos open parentheses 40 close parentheses equals 3 over straight H leads to straight H equals fraction numerator 3 over denominator cos open parentheses 40 close parentheses end fraction

  • 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 Tips and Tricks

  • SOHCAHTOA (like Pythagoras) can only be used in right-angled 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 x cm in the following triangle.

Give your answer to 3 significant figures.

cie-igcse-core-sohcahtoa---finding-lengths-rn-image

First label the triangle

Right Pointing Right Angled Triangle with measurements, IGCSE & GCSE Maths revision notes

We know A and we want to know O - that's TOA or tan space theta equals opposite over adjacent

tan open parentheses 43 close parentheses equals x over 9

Multiply both sides by 9

9 cross times tan open parentheses 43 close parentheses space equals space x

Enter on your calculator

x equals 8.3926...

Round to 3 significant figures

bold italic x bold equals bold 8 bold. bold 39 bold space bold cm

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