Trigonometry to Find Lengths (Edexcel IGCSE Maths)

SOHCAHTOA - Finding Lengths

Spec point — 4.8B

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

  • $\mathrm{sin}θ=\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}=\frac{O}{H}$
  • $\mathrm{cos}θ=\frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}=\frac{A}{H}$
  • $\mathrm{tan}θ=\frac{\mathrm{opposite}}{\mathrm{adjacent}}=\frac{O}{A}$

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

• 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 t rigonometric 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 ( 50 ) = A / 7 or cos ( 40 ) = 3 / H

• STEP 4 Rearrange and solve for the unknown letter

  • You will either need to multiply or divide

    • sin ( 50 ) = A / 7 leads to $\mathrm{A} = 7 \times \sin ( 50 )$
    • cos ( 40 ) = 3 / H leads to $\mathrm{H} = \frac{3}{\cos ( 40 )}$

• 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

Exam Hint

• 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.

Answer:

First label the triangle

We know A and we want to know O - that's TOA or $\mathrm{tan}θ=\frac{\mathrm{opposite}}{\mathrm{adjacent}}$

$\tan ( 43 ) = \frac{x}{9}$

Multiply both sides by 9

$9 \times \tan ( 43 ) = x$

Enter on your calculator

$x=8.3926...$

Round to 3 significant figures

$x=8.39\mathrm{cm}$