Right-Angled Trigonometry (AQA GCSE Further Maths): Revision Note

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

$a^2+b^2=c^2$

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:

$c= \sqrt{a^2+b^2}$

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

$a= \sqrt{c^2-b^2}$   or  $b= \sqrt{c^2-a^2}$

• 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

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

$\mathrm{Sin}\theta =\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}=\frac{O}{H}$

 $\mathrm{Cos}\theta =\frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}=\frac{\mathrm{A}}{\mathrm{H}}$

$\mathrm{Tan}\theta \mathrm{ }=\frac{\mathrm{opposite}}{\mathrm{adjacent}}=\frac{\mathrm{O}}{\mathrm{A}}$

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

• 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:

How can I 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 I 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:

 $\theta ={\mathrm{Sin}}^{-1}\frac{O}{H}$   ,    $\theta ={\mathrm{Cos}}^{-1}\frac{\mathrm{A}}{\mathrm{H}}$   ,   $\theta ={\mathrm{Tan}}^{-1}\frac{\mathrm{O}}{\mathrm{A}}$

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

Exact trig values

What are exact values in trigonometry?

• For certain angles the values of sin θ, cos θ and tan θ can be written exactly

  • This means using fractions and surds

  • You should be familiar with these values and be able to derive the values using geometry

• You are expected to know the exact values of sin, cos and tan for angles of 0°, 30°, 45°, 60°, 90°, 180° and their multiples

• The exact values you are expected to know are summarised here:

• Note that the values of sin θ  going from 0° to 90° match those of cos θ going from 90° to 0°

How are exact values in trigonometry derived?

• There are two special right-angled triangles that can be used to derive all of the exact values you need to know

• Consider a right-angles triangle with a hypotenuse of 2 units and a shorter side length of 1 unit

  • Using Pythagoras’ theorem the third side will be $\sqrt{3}$

  • The angles will be 90°, 60° and 30°

  • Using SOHCAHTOA gives…

    • Sin 60° = $\frac{\sqrt{3}}{2}$              Sin 30°  = $\frac{1}{2}$

    • Cos 60°  = $\frac{1}{2}$                 Cos 30° = $\frac{\sqrt{3}}{2}$

    • Tan 60°  = $\sqrt{3}$               Tan 30° = $\frac{1}{\sqrt{3}}$ = $\frac{\sqrt{3}}{3}$

• Consider an isosceles triangle with two equal side lengths (the opposite and adjacent) of 1 unit

  • Using Pythagoras’ theorem it will have a hypotenuse of $\sqrt{2}$

  • The two equal angles will be 45°

  • Using SOHCAHTOA gives…

    • Sin $45°=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$

    • Cos $45°=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$

    • Tan $45°$= 1