SOHCAHTOA (Cambridge (CIE) IGCSE International Maths)

Revision Note

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SOHCAHTOA

What is trigonometry?

  • Trigonometry is the mathematics of angles in triangles

  • It looks at the relationship between side lengths and angles of triangles

What are sin, cos and tan?

  • The three trigonometric functions sine, cosine and tangent

    • They come from ratios of side lengths in right-angled triangles

  • You must 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 often used to remember 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

  • 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

How can I use SOHCAHTOA to find missing angles?

  • STEP 1
    Label the sides of the triangle as H, O and A

  • STEP 2
    Identify which trigonometric ratio to use: sin, cos or tan

    • Write down the letters of the lengths you are given

    • Find the two letters in SOHCAHTOA to identify which ratio to use

      • If you have O and A then use tan

  • STEP 3
    Substitute the values into the relevant trigonometric formula

    • The angle will be unknown

      • tan open parentheses theta close parentheses equals 3 over 4 

  • STEP 4
    Substitute the fraction into the inverse trigonometric function

    • You normally need to press SHIFT on your calculator first

      • tan open parentheses theta close parentheses equals 3 over 4 leads to theta equals tan to the power of negative 1 end exponent open parentheses 3 over 4 close parentheses

  • STEP 5
    Type the expression into your calculator

    • The question might ask you to round your answer

    • If not then round to one decimal place

How do I find the shortest distance from a point to a line?

  • The shortest distance from any point to a line will always be the perpendicular distance

  • Form a right-angled triangle and then use SOHCAHTOA to find the relevant distance

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

Worked Example

Find the value of the angle y° in the following triangle.

Give your answer to 1 decimal place.

cie-igcse-core-sohcahtoa---finding-angles-image

First label the triangle

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

We know A and H - that's CAH or cos space theta equals adjacent over hypotenuse

cos open parentheses y close parentheses equals 8 over 23

Use inverse cos to find y

y equals cos to the power of negative 1 end exponent open parentheses 8 over 23 close parentheses

Enter on your calculator

y equals 69.6455...

Round to 1 decimal place

bold italic y bold equals bold 69 bold. bold 6 bold degree

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Dan Finlay

Author: Dan Finlay

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Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

Lucy Kirkham

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Lucy has been a passionate Maths teacher for over 12 years, teaching maths across the UK and abroad helping to engage, interest and develop confidence in the subject at all levels.Working as a Head of Department and then Director of Maths, Lucy has advised schools and academy trusts in both Scotland and the East Midlands, where her role was to support and coach teachers to improve Maths teaching for all.