Exact Trig Values (Cambridge International IGCSE Mathematics)

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 are expected to know the exact values of sin, cos and tan for

  • 0°, 30°, 45°, 60°, 90°, 180° and their multiples

How can I remember these exact trig values?

• Look at patterns in the table

  • Note the values of sin θ from 0° to 90° match cos θ in reverse, from 90° to 0°

Exam Hint Some people remember sin θ using the trick $\frac{\sqrt{0}}{2},\frac{\sqrt{1}}{2},\frac{\sqrt{2}}{2},\frac{\sqrt{3}}{2},\frac{\sqrt{4}}{2}$ which simplifies to $0,\frac{1}{2},\frac{\sqrt{2}}{2},\frac{\sqrt{3}}{2},1$.

• Two special right-angled triangles below can help you to find the exact values for 30º, 45º and 60º

  • Remember that by rationalising the denominator, $\frac{1}{\sqrt{2}}=\frac{1}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{2}}{2}$

Triangle for 45°

Triangle for 30° and 60°

• The trig graphs can help you to remember the exact values for 0º and multiples of 90º

How do I use exact trig values?

• You may come across trig questions in a non-calculator question

• In trig calculations, substitute in the exact trig values and solve as usual

  • E.g. Solve the equation $\mathrm{cos}45=\frac{x}{12}$

    • Replace $\mathrm{cos}45$ with $\frac{\sqrt{2}}{2}$ to give $\frac{\sqrt{2}}{2}=\frac{x}{12}$
    • Then you can solve for $x$

• On trig graphs, you may be expected to find a coordinate

  • E.g. The coordinates $( 30 , k )$ lie on the graph $y=\mathrm{tan}x$, find $k$

    • $k$ will be equal to $\mathrm{tan}30$
    • The exact value of $\mathrm{tan}30$ is $\frac{\sqrt{3}}{3}$
    • Therefore $k=\frac{\sqrt{3}}{3}$

Exam Hint

• Writing these out (or sketching the triangles/graphs) on your paper at the beginning of the exam means that you can use them as many times as you need to during the exam!

Worked Example Find the value of $x$ in the diagram below.

Give your answer as an exact value.

Answer:

Triangle ABC is a right-angled triangle, so use SOHCAHTOA

*We know the hypotenuse (AC) and we want to calculate the opposite (BC), so use *$\mathrm{sin}θ=\frac{O}{H}$

$\mathrm{sin}60=\frac{x}{28}$

Remember that $\mathrm{sin}60=\frac{\sqrt{3}}{2}$

So,

$\begin{gathered} \frac{ \sqrt{3} }{2} & = \frac{x}{28} \\ 28 \times \frac{ \sqrt{3} }{2} & = x \\ 14 \times \sqrt{3} & = x \end{gathered}$

Leave in exact (surd) form

Final answer: $x=14\sqrt{3}$cm