Right-Angled Trigonometry (Edexcel GCSE Maths: Foundation)

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
  • 
  •  
  • 

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 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
    •   or 
• STEP 4
  Rearrange and solve for the unknown letter
  • You will either need to multiply or divide
    • leads to 
    • leads to 
• 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
  • 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 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
    •  
• STEP 4
  Substitute the fraction into the inverse trigonometric function
  
  • You normally need to press SHIFT on your calculator first
    • leads to 
• 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

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