SOHCAHTOA (Cambridge International IGCSE Maths)

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

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

What is SOHCAHTOA?

• SOHCAHTOA is a mnemonic 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

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 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

    • $\sin ( 50 ) = \frac{O}{7}$ or cos ( 40 ) = 3 / H

• STEP 4 Rearrange and solve for the unknown letter

  • You will either need to multiply or divide

    • $\sin ( 50 ) = \frac{O}{7}$ leads to $O = 7 \times \sin ( 50 )$
    • 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

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

    • $\tan ( \\theta ) = \frac{3}{4}$

• STEP 4 Substitute the fraction into the inverse trigonometric function

  • You normally need to press SHIFT on your calculator first

    • $\tan ( \\theta ) = \frac{3}{4}$ leads to $\\theta = \tan ^{- 1} ( \frac{3}{4} )$

• 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

Exam Hint SOHCAHTOA (like Pythagoras) can only be used in right-angled triangles.

Exam Hint 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

$x=8.39\mathrm{cm}$

Worked Example Find the value of the angle $y$° in the following triangle.

Give your answer to 1 decimal place.

Answer:

First label the triangle

We know A and H - that's CAH or $\mathrm{cos}θ=\frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}$

$\cos ( y ) = \frac{8}{23}$

Use inverse cos to find $y$

$y = \cos ^{- 1} ( \frac{8}{23} )$

Enter on your calculator

$y=69.6455...$

Round to 1 decimal place

$y=69.6^{\circ}$