Sine and Cosine Rule

In this page, we will look at the Sine Rule, the Cosine Rule and the formula for the Area of a Triangle. The formula for these three are fairly straight forward to use, but don't take this topic too lightly, sometimes the exam questions can be quite challenging. Questions on this topic can even come up on the non-calculator paper when we use the angles 30°, 45°, 60°, ...


Key Concepts

On this page, you should learn about

  • the sine rule: \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)
  • the cosine rule: \(c^2=a^2+b^2-2ab\cos C\)
  • the area of a triangle =\({ 1\over 2}ab\sin C\)

Summary

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Test Yourself

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Exam-style Questions

Question 1

The following diagram shows a quadrilateral ABCD.

AB = 7cm , AD = 5cm ∠DAB=120° , ∠DBC=45° , ∠BCD=60°

BD = \(\sqrt{a}\)

CD = \(\sqrt{b}\)

\(a,b \in \mathbb{Z}\)

Find a and b

Hint

Full Solution

Question 2

The following diagram shows a quadrilateral ABCD.

AD = x – 1 , BD = x + 1 , DC = 2x and \(\angle CDA\) = 120°

The sum of the area of triangle ADC and triangle BDC is \(4 \sqrt{3}\)

Find x

Hint

Full Solution

Question 3

In a triangle ABC, AB = 8cm, BC = a, AC = b and \(\angle BAC\) = 30°

a) Show that \(b^2-8\sqrt{3}b+64-a^2=0\)

b) Hence find the possible values of a (in cm) for which the triangle has two possible solutions.

Hint

Full Solution

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