10.1 The Sine Rule

Posted on April 22, 2020 by user

10.1 The Sine Rule

In a triangle ABC in which the sides BC, CA and AB are denoted by a, b, and c as shown, and A, B, C are used to denote the angles at the vertices A, B, C respectively,

The sine rule can be used when

(i) two sides and one non-included angle or

(ii) two angles and one opposite side are given.

(A) If you know 2 angles and 1 side ⇒ Sine rule

Example:

Calculate the length, in cm, of AB.

Solution:

∠ACB = 180^o – (50^o + 70^o) = 60^o

$\begin{array}{l} \frac{A B}{\sin 60^{o}} = \frac{4}{\sin 50^{o}} \\ A B = \frac{4 \times \sin 60^{o}}{\sin 50^{o}} \\ A B = 4.522 cm \end{array}$

(B) If you know 2 sides and 1 angle (but not between them) ⇒ Sine rule

Example:

Calculate ∠ACB.

Solution:

$\begin{array}{l} \frac{28}{\sin 54^{o}} = \frac{26}{\sin ∠ A C B} \\ \sin ∠ A C B = \frac{26 \times \sin 54^{o}}{28} \\ \sin ∠ A C B = 0.7512 \\ ∠ A C B = {48.7}^{o} \end{array}$

(C) Case of ambiguity (2 possible triangles)

Example

Calculate ∠ACB, θ.

Solution:

Two possible triangle with these measurement

AB = 26cm BC = 28 cm Ð BAC = 54^o

$\begin{array}{l} \frac{26}{\sin θ} = \frac{28}{\sin 54^{o}} \\ \sin θ = 0.7512 \\ θ = \sin^{- 1} 0.7512 \\ θ = {48.7}^{o}, 180^{o} - {48.7}^{o} \\ θ = {48.7}^{o} (Acute angle), {131.3}^{o} (Obtuse angle) \end{array}$