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

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

  • 

  • 

  • 

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 

• STEP 4
  Rearrange and solve for the unknown letter

  • You will either need to multiply or divide

    • leads to 

    • leads to 

• 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