The cosine rule

Summary

In this tutorial, the cosine rule is explained with two practical examples. The first shows how to use the formula to find a missing side of a triangle when two sides and an included angle are known. The second demonstrates how to calculate a missing angle when all three sides of a triangle are provided. The formula, a² = b² + c² - 2bc cos(A), is broken down step-by-step, emphasizing the relationship between the sides and angles of a triangle. This method is vital for solving real-world problems involving triangles.

Takeaways

• 😀 The Cosine Rule is a formula used to solve triangles when two sides and the included angle or all three sides are known.
• 😀 The formula for the Cosine Rule is: a² = b² + c² - 2bc * cos(A), where a, b, and c are sides and A, B, and C are the respective opposite angles.
• 😀 The sides and angles are labeled in correspondence, meaning side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C.
• 😀 The Cosine Rule can be used to find a missing side when two sides and the included angle are known.
• 😀 The Cosine Rule can also be used to find a missing angle when all three sides of the triangle are given.
• 😀 In the first example, the missing side is calculated by using the Cosine Rule with a 100° angle and side lengths of 6.5 cm and 8.7 cm, resulting in a side length of approximately 11.7 cm.
• 😀 In the second example, with side lengths 10.6 cm, 8.2 cm, and 5.9 cm, the missing angle is found by using the Cosine Rule, yielding an angle of approximately 96.1°.
• 😀 When applying the Cosine Rule to find a missing side, the formula involves squaring the known sides, subtracting the product of the sides and cosine of the angle, and then taking the square root.
• 😀 When finding a missing angle, you first use the Cosine Rule, simplify the equation, and then apply the inverse cosine to find the angle.
• 😀 The Cosine Rule is crucial for solving non-right-angled triangles where the Pythagorean theorem cannot be used.

Q & A

What is the cosine rule?

The cosine rule is a formula used in trigonometry to find a missing side or angle in a triangle. It states that: a² = b² + c² - 2bc * cos(A), where a, b, and c are the sides of the triangle, and A is the angle opposite side 'a'.

How are the sides and angles labeled in the cosine rule formula?

In the cosine rule, the sides are labeled as 'a', 'b', and 'c', where 'a' is opposite the angle A, 'b' is opposite angle B, and 'c' is opposite angle C. This helps to correctly associate the sides and angles.

What information is needed to use the cosine rule to find a missing side?

To use the cosine rule to find a missing side, you need the lengths of the other two sides and the included angle (the angle between the two known sides).

What was the first example in the video about?

The first example involved using the cosine rule to find the length of the hypotenuse in a triangle, where one angle was 100 degrees and the lengths of the other two sides were 6.5 cm and 8.7 cm.

How do you apply the cosine rule to find a missing side, as demonstrated in the first example?

To find the missing side (a), plug the known values into the cosine rule formula: a² = b² + c² - 2bc * cos(A), and solve for 'a'. The result is approximately 11.7 cm.

What is the second example about?

The second example demonstrates how to use the cosine rule to find a missing angle in a triangle. The sides of the triangle were 8.2 cm, 5.9 cm, and 10.6 cm, and the goal was to find the missing angle A.

How do you apply the cosine rule to find a missing angle, as shown in the second example?

To find the missing angle A, first assign the sides to 'a', 'b', and 'c'. Then, use the cosine rule formula, solve for cos(A), and find the angle by taking the inverse cosine (cos⁻¹) of the result. In this case, the missing angle A is approximately 96.1 degrees.

What is the process of solving for a missing angle using the cosine rule?

First, assign the sides to 'a', 'b', and 'c'. Then, plug these into the cosine rule formula, simplify the equation to solve for cos(A), and finally, use the inverse cosine function to find the angle A.

Why is it important to correctly label the sides and angles in the cosine rule formula?

Correctly labeling the sides and angles is crucial because the formula relies on the relationship between the sides and their opposite angles. Incorrect labeling could lead to wrong results.

Can the cosine rule be used to find an angle when only two sides and the included angle are known?

Yes, the cosine rule can be used to find an angle when two sides and the included angle are known. By rearranging the formula, you can solve for the cosine of the angle and then use the inverse cosine function to find the angle.