How to Find an Unknown Angle in a Triangle Given Sides and Area

When you are given two sides of a triangle along with the area, finding the unknown angle can be an intriguing and insightful problem. This article will guide you through the process with detailed steps and examples.

Understanding the Relationship Between Triangle Area, Sides, and Angles

The area of a triangle is given by the formula: Area 1/2 * a * b * sin(θ). Here, a and b are the lengths of the two sides, and θ is the angle between them. By rearranging this formula, you can solve for the sine of the angle θ:

sin(θ) 2 * Area / (a * b)

This equation allows you to determine the sine of the angle, which is a crucial step in solving the problem. Let's walk through an example to clarify the process.

Example Walkthrough

Consider a triangle with the following dimensions: two sides a 12 cm and b 15 cm, and an area of A 89.66 cm2. We will find the unknown angles step by step.

Step 1: Finding Angle C

The area of the triangle is given by:

A 1/2 * a * b * sin(C)

Substituting the known values:

89.66 1/2 * 12 * 15 * sin(C)

Solving for sin(C):

sin(C) 2 * 89.66 / (12 * 15) 0.996222222

From this, we can find the angle C using the arcsin function:

C arcsin(0.996222222) 85.04812701° ≈ 85.05°

Since C is the largest angle in the triangle and is opposite the longest side (c 20 cm), it must be the largest angle.

Step 2: Finding Angle A

Using the Law of Sines, we can find angle A:

A arcsin(2 * A / (b * c)) arcsin(2 * 89.66 / (15 * 20)) arcsin(0.597733333) 36.7077315° ≈ 36.71°

This step involves using the Law of Sines, which states that the ratio of the length of a side to the sine of its opposite angle is the same for all three sides of the triangle.

Step 3: Finding Angle B

The third angle B can be found by subtracting the sum of angles A and C from 180° (since the sum of the internal angles of a triangle is always 180°):

B 180° - A - C 180° - 36.71° - 85.05° 58.24° ≈ 48.34°

This method uses the fact that the sum of the angles in a triangle is 180 degrees.

Conclusion and Further Reading

The problem of finding an unknown angle in a triangle given sides and area involves a combination of trigonometric and geometric principles. By using the sine formula for calculating the area and the Law of Sines, you can effectively solve this type of problem.

Remember, all triangles are ultimately comprised of right triangles, and understanding this can help solve even more complex problems. If you need further assistance or explore other mathematical concepts, don't hesitate to delve into resources such as trigonometry books or online tutorials.