Laws of Sines and Cosines

Introduction

In trigonometry, the laws of sines and cosines are fundamental to solving problems involving triangles. These rules allow us to find unknown angles and sides of a triangle when certain other elements of the triangle are known. They are essential tools not only in geometry but also in real-world applications such as engineering, navigation, and physics.

Law of sines

The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of the angle opposite it is constant. It can be written as:

a / sin(A) = b / sin(B) = c / sin(C)

Here, a, b and c are the lengths of the sides of the triangle respectively, and A, B and C are opposite angles.

Visual example of the Law of Sines

In the above triangle, you can apply the Law of Sines to find the unknown sides or angles.

Example application of the Law of Sines

Suppose you have a triangle with sides a = 8, angle A = 30°, and angle B = 45°. To find side b, use:

a / sin(A) = b / sin(B)

First, find the sine value:

sin(30°) = 0.5 and sin(45°) = 0.707

Use of the Law of Sines:

8 / 0.5 = b / 0.707

Solving this gives:

b = 8 * 0.707 / 0.5 = 11.312

Thus, the length of side b is approximately 11.312.

Law of cosines

The Law of Cosines is useful for calculating the unknown side of a triangle when two sides and their included angle are known, or for finding an angle when all three sides are known. The formula is:

c² = a² + b² - 2ab * cos(C)

This can be rearranged to find the angle:

cos(C) = (a² + b² - c²) / (2ab)

Visual example of the Law of Cosines

In the triangle shown, the Law of Cosines can be used to find unknown angles or sides.

Example application of the Law of Cosines

Imagine a triangle with sides a = 5, b = 7 and angle C = 60°. To find the third side c, use:

c² = a² + b² - 2ab * cos(C)

Start by finding cos(60°):

cos(60°) = 0.5

Substitute and solve:

c² = 5² + 7² - 2 * 5 * 7 * 0.5 c² = 25 + 49 - 35 c² = 39 c = sqrt(39) ≈ 6.24

Therefore, the length of side c is approximately 6.24.

Use of both laws together

Sometimes, you may need to use both the Law of Sines and the Law of Cosines to solve a problem. For example, if you have a triangle with two sides and no angles between them, you can first use the Law of Cosines to find the third side and then use the Law of Sines to find the unknown angles.

Example combining both laws

Imagine a triangle with sides a = 9, b = 12 and B = 75°. To calculate angle A and side c, first find c using the law of cosines:

c² = a² + b² - 2ab * cos(B) c² = 9² + 12² - 2 * 9 * 12 * cos(75°)

The cosine of the angle 75° is:

cos(75°) ≈ 0.2588

Substitute into the formula:

c² = 81 + 144 - 216 * 0.2588 c² = 225 - 55.9408 c² ≈ 169.0592 c ≈ sqrt(169.0592) c ≈ 13

Then, use the law of sines to find angle A:

sin(A) / a = sin(B) / b sin(A) / 9 = sin(75°) / 12

Find sin(75°):

sin(75°) ≈ 0.9659

Solve for sin(A):

sin(A) = 9 * 0.9659 / 12 sin(A) ≈ 0.7244

Finally determine the angle A:

A ≈ arcsin(0.7244) ≈ 46.28°

Therefore, angle A is approximately 46.28°, and side c is approximately 13.