How to Put Cosine Rule Into Calculator

The cosine rule is a fundamental trigonometric formula used to find the length of a side of a triangle when you know the lengths of the other two sides and the included angle. This guide explains how to apply the cosine rule using a calculator, including step-by-step instructions and practical examples.

What is the Cosine Rule?

The cosine rule, also known as the law of cosines, is a relationship between the lengths of the sides of a triangle and the cosine of one of its angles. It extends the Pythagorean theorem to non-right-angled triangles.

Cosine Rule Formula:

c² = a² + b² - 2ab cos(C)

Where:

c = length of the side opposite angle C
a and b = lengths of the other two sides
C = included angle between sides a and b

The cosine rule is particularly useful in surveying, navigation, and engineering where you need to calculate distances between points when you know the angles and some side lengths.

How to Use the Cosine Rule

To use the cosine rule effectively, follow these steps:

Identify the known values: two sides of the triangle and the included angle.
Write down the cosine rule formula: c² = a² + b² - 2ab cos(C).
Substitute the known values into the formula.
Calculate the value of c using a calculator.
Interpret the result in the context of your problem.

Note: Ensure all angle measurements are in the same units (degrees or radians) and that the angle is between the two known sides.

Using a Calculator for the Cosine Rule

Most scientific calculators have a built-in cosine function that can be used to apply the cosine rule. Here's how to use a calculator for this purpose:

Enter the values of sides a and b.
Square both values (a² and b²).
Calculate the product of a, b, and the cosine of angle C (2ab cos(C)).
Subtract the third term from the sum of the first two terms (a² + b² - 2ab cos(C)).
Take the square root of the result to find the length of side c.

For more complex calculations, you may need to use a programming calculator or software that supports trigonometric functions.

Example Calculation

Let's solve a sample problem using the cosine rule:

Problem: In triangle ABC, side AB = 5 cm, side AC = 7 cm, and angle A = 45°. Find the length of side BC.

Solution:

Identify the known values: a = 5 cm, b = 7 cm, angle C = 45°.
Apply the cosine rule: c² = a² + b² - 2ab cos(C).
Calculate each term:
- a² = 5² = 25
- b² = 7² = 49
- 2ab cos(C) = 2 × 5 × 7 × cos(45°) ≈ 70 × 0.7071 ≈ 49.497
Combine the terms: c² = 25 + 49 - 49.497 ≈ 24.503.
Take the square root: c ≈ √24.503 ≈ 4.95 cm.

The length of side BC is approximately 4.95 cm.

Frequently Asked Questions

What is the difference between the cosine rule and the Pythagorean theorem?: The Pythagorean theorem applies only to right-angled triangles, while the cosine rule works for any triangle, including right-angled ones.
When should I use the cosine rule instead of the sine rule?: Use the cosine rule when you know two sides and the included angle, and need to find the third side. Use the sine rule when you know two angles and one side, or two sides and a non-included angle.
Can I use the cosine rule for triangles with angles greater than 90°?: Yes, the cosine rule works for all types of triangles, including obtuse-angled triangles.
What if I don't have a calculator with trigonometric functions?: You can use online calculators or programming software to perform the calculations.
How accurate are the results from the cosine rule?: The accuracy depends on the precision of the input values and the calculator used. For most practical purposes, the results are sufficiently accurate.