Trigonometry III

Trigonometry – Sine and Cosine Rules

Lesson Objectives

Understand and apply the sine rule for any triangle.
Understand and apply the cosine rule for any triangle.
Solve problems involving non-right-angled triangles.

Lesson Introduction

When dealing with non-right-angled triangles, we use the Sine Rule and the Cosine Rule to find unknown sides or angles.

Sine Rule

The sine rule states:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Used when: we know either two angles and one side (AAS/ASA), or two sides and a non-included angle (SSA).

Cosine Rule

The cosine rule states:

$c^2 = a^2 + b^2 - 2ab\cos C$

Used when: we know two sides and the included angle (SAS), or all three sides (SSS).

Worked Example

Sine Rule Examples

Example 1:
In triangle ABC,

A = 50^\circ, B = 60^\circ, a = 7

. Find side

b

\frac{7}{\sin(50^\circ)} = \frac{b}{\sin(60^\circ)} \Rightarrow b = \frac{7 \cdot \sin(60^\circ)}{\sin(50^\circ)} \approx 7.84

Example 2:
Given

a = 10, A = 30^\circ, B = 45^\circ

, find

b

b = \frac{10 \cdot \sin(45^\circ)}{\sin(30^\circ)} \approx 14.14

Example 3:
Find angle

C

a = 6, c = 10, A = 40^\circ

\frac{6}{\sin(40^\circ)} = \frac{10}{\sin(C)} \Rightarrow \sin(C) = \frac{10 \cdot \sin(40^\circ)}{6} \approx 1.071 \Rightarrow \text{Not possible (no triangle)}

Example 4:
Given triangle DEF:

D = 70^\circ, F = 60^\circ, d = 12

. Find side

f

E = 180^\circ - 70^\circ - 60^\circ = 50^\circ

\frac{12}{\sin(70^\circ)} = \frac{f}{\sin(60^\circ)} \Rightarrow f \approx 11.3

Example 5:

a = 15, A = 65^\circ, B = 80^\circ

. Find side

b

b = \frac{15 \cdot \sin(80^\circ)}{\sin(65^\circ)} \approx 16.4

Cosine Rule Examples

Example 1:
Given

a = 5, b = 7, C = 60^\circ

. Find side

c

c^2 = 5^2 + 7^2 - 2(5)(7)\cos(60^\circ) = 25 + 49 - 70(0.5) = 39 \Rightarrow c = \sqrt{39} \approx 6.24

Example 2:
In triangle PQR,

p = 10, q = 14, R = 90^\circ

. Find side

r

r^2 = 10^2 + 14^2 - 2(10)(14)\cos(90^\circ) = 100 + 196 = 296 \Rightarrow r \approx 17.2

Example 3:
Find angle

C

a = 9, b = 8, c = 10

\cos C = \frac{9^2 + 8^2 - 10^2}{2(9)(8)} = \frac{145 - 100}{144} = \frac{45}{144} \approx 0.3125

C \approx \cos^{-1}(0.3125) \approx 71.8^\circ

Example 4:
Triangle XYZ has sides

x = 11, y = 14, z = 13

. Find angle

Z

\cos Z = \frac{11^2 + 14^2 - 13^2}{2(11)(14)} \approx 0.378 \Rightarrow Z \approx 67.8^\circ

Example 5:
[WAEC] Find angle

B

a = 12, b = 9, c = 10

. (Past Question)

\cos B = \frac{12^2 + 10^2 - 9^2}{2(12)(10)} = \frac{144 + 100 - 81}{240} = \frac{163}{240} \approx 0.679 \Rightarrow B \approx 47.3^\circ

Exercises

Given $a = 5, A = 50^\circ, B = 60^\circ$ , find side b.
In triangle XYZ, $x = 10, y = 12, X = 30^\circ$ . Find angle Y.
[NECO] Triangle has $a = 11, b = 13, C = 60^\circ$ . Find c. (Past Question)
Given sides 7, 8, and 9, find the angle opposite the side of length 9.
Find side c if $a = 9, b = 10, C = 45^\circ$ .
[WAEC] Triangle has $A = 40^\circ, a = 8, B = 60^\circ$ . Find b. (Past Question)
In triangle PQR, $P = 55^\circ, Q = 65^\circ, p = 9$ . Find side q.
Given $a = 7, b = 10, c = 11$ , find angle C.
[JAMB] Solve for angle C if $a = 6, b = 6, c = 6$ . (Past Question)
Given $a = 9, b = 12, C = 100^\circ$ , find side c.

Conclusion/Recap

The sine and cosine rules are powerful tools for solving problems involving non-right-angled triangles. Use the sine rule when dealing with angle-side ratios, and the cosine rule when you have side-angle-side or all three sides.