DP IB Maths: AA HL

Revision Notes

3.3.1 Pythagoras & Right-Angled Trigonometry

Test Yourself

Pythagoras

What is the Pythagorean theorem?

  • Pythagoras’ theorem is a formula that works for right-angled triangles only
  • It states that for any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides
    • The hypotenuse is the longest side in a right-angled triangle
      • It will always be opposite the right angle
    • If we label the hypotenuse c, and label the other two sides a and b, then Pythagoras’ theorem tells us that

begin mathsize 22px style a squared space plus space b squared space equals space c squared end style

  • The formula for Pythagoras’ theorem is assumed prior knowledge and is not in the formula booklet
    • You will need to remember it

How can we use Pythagoras’ theorem?

  • If you know two sides of any right-angled triangle you can use Pythagoras’ theorem to find the length of the third side
    • Substitute the values you have into the formula and either solve or rearrange
  • To find the length of the hypotenuse you can use:

begin mathsize 22px style c equals blank square root of a squared plus b squared end root end style

  • To find the length of one of the other sides you can use:

a space equals blank square root of c squared space minus space b squared end root   or  b space equals blank square root of c squared space minus space a squared end root

  • Note that when finding the hypotenuse you should add inside the square root and when finding one of the other sides you should subtract inside the square root
  • Always check your answer carefully to make sure that the hypotenuse is the longest side
  • Note that Pythagoras’ theorem questions will rarely be standalone questions and will often be ‘hidden’ in other geometry questions

What is the converse of the Pythagorean theorem?

  • The converse of the Pythagorean theorem states that if a squared space plus space b squared space equals space c squared  is true then the triangle must be a right-angled triangle
    • This is a very useful way of determining whether a triangle is right-angled
  • If a diagram in a question does not clearly show that something is right-angled, you may need to use Pythagoras’ theorem to check

Exam Tip

  • Pythagoras' theorem pops up in lots of exam questions so bear it in mind whenever you see a right-angled triangle in an exam question!

Worked example

ABCDEF is a chocolate bar in the shape of a triangular prism.  The end of the chocolate bar is an isosceles triangle where AC = 3 cm and AB = BC = 5 cm.  M is the midpoint of AC. This information is shown in the diagram below.

diagram-for-we-3-3-1-pythag

Calculate the length BM.

3-3-1-ai-sl-pythag-we-solution

Right-Angled Trigonometry

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

    • These are not in the formula book, you must remember them
  • The mnemonic SOHCAHTOA 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

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

Exam Tip

  • You need to remember the sides involved in the different trig ratios as they are not given to you in the exam

Worked example

Find the values of x and y in the following diagram. Give your answers to 3 significant figures.

sa-diagram-for-we-3-3-1-trig

3-3-1-ai-sl-r-a-trig-we-solution

3D Problems

How does Pythagoras work in 3D?

  • 3D shapes can often be broken down into several 2D shapes
  • With Pythagoras’ Theorem you will be specifically looking for right-angled triangles
    • The right-angled triangles you need will have two known sides and one unknown side
    • Look for perpendicular lines to help you spot right-angled triangles
  • There is a 3D version of the Pythagorean theorem formula:

begin mathsize 22px style d squared space equals space x squared space plus space y squared space plus space z squared end style 

    • However it is usually easier to see a problem by breaking it down into two or more 2D problems

How does SOHCAHTOA work in 3D?

  • Again look for a combination of right-angled triangles that would lead to the missing angle or side
  • The angle you are working with can be awkward in 3D
    • The angle between a line and a plane is not always obvious
    • If unsure put a point on the line and draw a new line to the plane
      • This should create a right-angled triangle

 

3DPythagTrig Notes fig6

Exam Tip

  • Annotate diagrams that are given to you with values that you have calculated
  • It can be useful to make additional sketches of parts of any diagrams that are given to you, especially if there are multiple lengths/angles that you are asked to find
  • If you are not given a diagram, sketch a nice, big, clear one!

Worked example

A pencil is being put into a cuboid shaped box which has dimensions 3 cm by 4 cm by 6 cm. Find:

 

a)
the length of the longest pencil that could fit inside the box,

ai-sl-3-3-1-3d-pythag-trig-we-solution-a

 

b)
the angle that the pencil would make with the top of the box.

ai-sl-3-3-1-3d-pythag-trig-we-solution-b

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Amber

Author: Amber

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.