Right-Angled Trigonometry (AQA GCSE Maths)

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

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

• 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:

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 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:

    ,      ,   

• After choosing the correct ratio and substituting the values use the inverse trigonometric functions on your calculator to find the correct answer

What are the angles of elevation and depression?

• If a person looks at an object that is not on the same horizontal line as their eye-level they will be looking at either an angle of elevation or depression
  • If a person looks up at an object their line of sight will be at an angle of elevation with the horizontal
  • If a person looks down at an object their line of sight will be at an angle of depression with the horizontal
• Angles of elevation and depression are measured from the horizontal
• Right-angled trigonometry can be used to find an angle of elevation or depression or a missing distance
• Tan is often used in real-life scenarios with angles of elevation and depression
  • For example if we know the distance we are standing from a tree and the angle of elevation of the top of the tree we can use Tan to find its height
  • Or if we are looking at a boat at to sea and we know our height above sea level and the angle of depression we can find how far away the boat is