Syllabus Edition

First teaching 2023

First exams 2025

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SOHCAHTOA (CIE IGCSE Maths: Extended)

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
  • 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 space theta blank equals space opposite over hypotenuse space equals space O over 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

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 or knowledge of trig exact values 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:

 theta space equals space Sin to the power of negative 1 end exponent space O over H   ,    theta space equals space Cos to the power of negative 1 end exponent space straight A over straight H   ,   theta space equals space Tan to the power of negative 1 end exponent space straight O over straight A

  • After choosing the correct ratio and substituting the values use the inverse trigonometric functions on your calculator or your knowledge of trig exact values to find the correct answer

Do sin, cos and tan work with obtuse angles?

  • Yes, your calculator or knowledge of trig exact values can be used to find sin, cos and tan of any angle
  • Some patterns can occur that will help if you need to find an obtuse angle 
    • sin(x) = sin(180° - x)
      • For example, sin(150°) = sin(180° - 150°) = sin(30°)
    • cos(x) = -cos(180 - x)
      • For example, cos(150°) = -cos(180° - 150°) = -cos(30°)
    • tan(x) = -tan(180 - x)
      • For example, tan(150°) = -tan(180° - 150°) = -tan(30°)
  • Be careful if a question requires you to find the size of an obtuse angle, you calculator will give you the acute angle so use one of the rules above to find the obtuse angle

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 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
  • In the exam, 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.

Two Right Angled Triangles with measurements, IGCSE & GCSE Maths revision notes

To find x, 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 tanθ 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

To find y, 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 cosθ 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 3 significant figures

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

Worked example

In the following diagram:
A B space equals space 12 space cm
A C is a straight line, with A D space equals space 9 space cm and A C space equals space 22 space cm

Back to Back Right Angled Triangles, IGCSE & GCSE Maths revision notes

Find the shortest distance from D to the side B C. Give your answer to 1 decimal place.

The shortest distance will be the perpendicular line directly from D to B C.

We first need to find the length of B D using triangle A B D

Note that B D is a shorter side

B D space equals space square root of 12 squared minus 9 squared end root space equals space square root of 63 space equals space 7.93725...

It is best to leave rounding until the very end, use square root of 63 in subsequent working

Now we can find angle B C D using SOHCAHTOA with triangle B C D

D C space equals space 22 space minus space 9 space equals space 13 space cm

angle space B C D space equals tan to the power of negative 1 end exponent open parentheses fraction numerator square root of 63 over denominator 13 end fraction close parentheses space equals space 31.4064... degree

CIUQBgW2_cie-2025-igcse-sohcahtoa-we2

Use right-angled trigonometry (SOHCAHTOA) to find the length of the line D X.

The hypotenuse is known and the side opposite the known angle is needed so use  sin space theta space equals straight O over straight H.

sin space 31.41 space equals fraction numerator space D X over denominator 13 end fraction

Rearrange to find D X.

table row cell D X space end cell equals cell 13 space sin space 31.41 degree end cell row blank equals cell space 6.775061... end cell end table

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Amber

Author: Amber

Expertise: Maths

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.