Right-Angled Trigonometry (Edexcel GCSE Maths)

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Naomi C

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Naomi C

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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
  • 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

Sin space theta blank equals space opposite over hypotenuse space equals space O over 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

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

Right-Angled-Triangles-OAH-Theta, IGCSE & GCSE Maths revision notes

  • 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:

Right-Angled Triangles Diagram 1

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:

 theta space equals space Sin to the power of negative 1 end exponent space O over H   ,    theta space equals space Cos to the power of negative 1 end exponent space straight A over straight H   ,   theta space equals space Tan to the power of negative 1 end exponent space straight O over straight A

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

Examiner Tip

  • SOHCAHTOA (like Pythagoras) can only be used in right-angles triangles – for triangles that are not right-angled, you will need to use the Sine Rule or the Cosine Rule
  • Also, make sure your calculator is set to measure angles in degrees

Worked example

Find the values of x and y in the following triangles.

Give your answers to 3 significant figures.

Two Right Angled Triangles with measurements, IGCSE & GCSE Maths revision notes

To find x, 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 tanθ 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

To find y, 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 cosθ 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 3 significant figures

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
  • 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

IV60s58R_ib-ai-sl-3-3-3-applications-of-trigonometry-diagram-1

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

A cliff is perpendicular to the sea and the top of the cliff stands 24 m above the level of the sea. The angle of depression from the cliff to a boat at sea is 35°. At a point xm up the cliff is a flag marker and the angle of elevation from the boat to the flag marker is 18°.

a)
Draw and label a diagram to show the top of the cliff, T, the foot of the cliff, F, the flag marker, M, and the boat, B, labelling all the angles and distances given above.

 3-3-3-ai-sl-elevation--depression-we-solution-i

b)
Find the distance from the boat to the foot of the cliff.

3-3-3-ai-sl-elevation--depression-we-solution-ii

c)
Find the value of x.

3-3-3-ai-sl-elevation--depression-we-solution-iii

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Naomi C

Author: Naomi C

Expertise: Maths

Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.