Pythagoras & Right-Angled Trigonometry (DP IB Maths: AI HL)

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

• 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:

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

  or  

• 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  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

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

• • 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

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

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:

• • 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