Finding the Length of the Shortest Side of a Right Triangle Using Trigonometry

Have you ever come across a situation where you need to determine the length of the shortest side of a right-angle triangle, given the hypotenuse and the angle between the shortest side and the hypotenuse? This is a common problem in trigonometry, and it can be solved by using trigonometric relationships. Let’s explore a step-by-step method to find the solution to the problem: 'What is the length of the shortest side of a right-angle triangle with the hypotenuse being 10 cm and the angle between the shortest side and hypotenuse being 20 degrees?'

Understanding the Problem

We are given the hypotenuse (c) as 10 cm and the angle (theta) between the shortest side (adjacent) and the hypotenuse as 20 degrees. The goal is to find the length of the shortest side, which is adjacent to the angle.

Trigonometric Relationships

The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse. Mathematically, this is represented as:

cos(theta) (adjacent side) / hypotenuse

We can rearrange this formula to solve for the adjacent side:

adjacent side hypotenuse * cos(theta)

Solution Steps

Identify the given values:

Use the cosine relationship to solve for the adjacent side:

adjacent side 10 * cos(20 degrees)

Calculate the cosine of 20 degrees. Using a calculator, we find:

cos(20 degrees) ≈ 0.9397

Substitute the values into the equation:

adjacent side 10 * 0.9397 ≈ 9.397 cm

Conclusion

The length of the shortest side of the right triangle is approximately 9.40 cm. This method can be used to solve similar trigonometric problems where you need to find the length of a side given the hypotenuse and an angle.

Additional Information

While the example above focuses on the cosine relationship, it's also worth noting the sine and tangent relationships in a right triangle. These relationships are:

The mnemonic SOHCAHTOA can help you remember these relationships: Sine Opposite over Hypotenuse, Cosine Adjacent over Hypotenuse, Tangent Opposite over Adjacent.