Right-Angled Trigonometry (Edexcel GCSE Maths: Foundation)

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SOHCAHTOA - Finding Lengths

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
    • sin space theta blank equals space opposite over hypotenuse space equals space straight O over straight H
    •  cos space theta blank equals space adjacent over hypotenuse space equals space straight A over straight H
    • tan space theta blank equals space opposite over adjacent space equals space straight O over straight A

Trigonometric formulae for a right-angled triangle

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

Labelling a right-angled triangle using H, O and A

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
      • sin open parentheses 50 close parentheses equals straight A over 7  or  cos open parentheses 40 close parentheses equals 3 over straight H
  • STEP 4
    Rearrange and solve for the unknown letter
    • You will either need to multiply or divide
      • sin open parentheses 50 close parentheses equals straight A over 7 leads to straight A equals 7 cross times sin open parentheses 50 close parentheses
      • cos open parentheses 40 close parentheses equals 3 over straight H leads to straight H equals fraction numerator 3 over denominator cos open parentheses 40 close parentheses end fraction
  • 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

Examiner Tip

  • SOHCAHTOA (like Pythagoras) can only be used in right-angles triangles
  • 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.

cie-igcse-core-sohcahtoa---finding-lengths-rn-image

First label the triangle

Right Pointing Right Angled Triangle with measurements, IGCSE & GCSE Maths revision notes

We know A and we want to know O - that's TOA or tan space theta equals opposite over adjacent

tan open parentheses 43 close parentheses equals x over 9

Multiply both sides by 9

9 cross times tan open parentheses 43 close parentheses space equals space x

Enter on your calculator

x equals 8.3926...

Round to 3 significant figures

bold italic x bold equals bold 8 bold. bold 39 bold space bold cm

SOHCAHTOA - Finding Angles

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
      • tan open parentheses theta close parentheses equals 3 over 4 
  • STEP 4
    Substitute the fraction into the inverse trigonometric function
    • You normally need to press SHIFT on your calculator first
      • tan open parentheses theta close parentheses equals 3 over 4 leads to theta equals tan to the power of negative 1 end exponent open parentheses 3 over 4 close parentheses
  • 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

Worked example

Find the value of the angle y° in the following triangle.

Give your answer to 1 decimal place.

cie-igcse-core-sohcahtoa---finding-angles-image

First label the triangle

Left Pointing Right Angled Triangle with measurements, IGCSE & GCSE Maths revision notes

We know A and H - that's CAH or cos space theta equals adjacent over hypotenuse

cos open parentheses y close parentheses equals 8 over 23

Use inverse cos to find y

y equals cos to the power of negative 1 end exponent open parentheses 8 over 23 close parentheses

Enter on your calculator

y equals 69.6455...

Round to 1 decimal place

bold italic y bold equals bold 69 bold. bold 6 bold degree

Elevation & Depression

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

Angles of elevation and depression

Examiner Tip

  • It may be useful to draw more than one diagram if the triangles that you are interested in overlap one another

Worked example

The diagram below shows a police helicopter using a high beam light to find someone at point A on the ground. The distance between the helicopter and point A is 22 metres. The angle of depression from the helicopter to point A is 30°.

angle-of-depression-we

Find the vertical height of the helicopter above the ground.

Draw a right-angled triangle

angle-of-depression-we-solution

You have the hypotenuse and you want to find the opposite side to the angle
Use SOH

sin open parentheses 30 close parentheses equals h over 22

Multiply both sides by 22

h equals 22 cross times sin open parentheses 30 close parentheses
h equals 11

Height is 22 metres

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Dan

Author: Dan

Expertise: Maths

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.