SOHCAHTOA

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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

  • $\sin \theta = \frac{\mathrm{opposite}}{\mathrm{hypotenuse}} = \frac{\mathrm{O}}{\mathrm{H}}$

  • $\cos \theta = \frac{\mathrm{adjacent}}{\mathrm{hypotenuse}} = \frac{\mathrm{A}}{\mathrm{H}}$

  • $\tan \theta \mathrm{ }= \frac{\mathrm{opposite}}{\mathrm{adjacent}} = \frac{\mathrm{O}}{\mathrm{A}}$

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 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

    • $\sin \left(50\right)=\frac{\mathrm{A}}{7}$  or  $\cos \left(40\right)=\frac{3}{\mathrm{H}}$

• STEP 4
  Rearrange and solve for the unknown letter

  • You will either need to multiply or divide

    • $\sin \left(50\right)=\frac{\mathrm{A}}{7}$ leads to $\mathrm{A}=7\times \sin \left(50\right)$

    • $\cos \left(40\right)=\frac{3}{\mathrm{H}}$ leads to $\mathrm{H}=\frac{3}{\cos \left(40\right)}$

• 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

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

• STEP 4
  Substitute the fraction into the inverse trigonometric function

  • You normally need to press SHIFT on your calculator first

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

• 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