Right-Angled Trigonometry (CIE IGCSE Maths: Extended)

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
  

Do sin, cos and tan work with obtuse angles?

• Yes, your calculator can be used to find sin, cos and tan of any angle
• Some patterns can occur that will help if you need to find an obtuse angle 
  • sin(x) = sin(180° - x)
    
    • For example, sin(150°) = sin(180° - 150°) = sin(30°)
  • cos(x) = -cos(180 - x)
    • For example, cos(150°) = -cos(180° - 150°) = -cos(30°)
  • tan(x) = -tan(180 - x)
    • For example, tan(150°) = -tan(180° - 150°) = -tan(30°)
• Be careful if a question requires you to find the size of an obtuse angle, you calculator will give you the acute angle so use one of the rules above to find the obtuse angle

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