Solving a Right Triangle with Given Sides: A Step-by-Step Guide

When given the sides of a right triangle, it's quite straightforward to find the angles using the Pythagorean theorem and trigonometric functions. Let's walk through the solution for a triangle ABC with sides a 24, b 7, and c 25.

Step 1: Verify the Right Triangle

To ensure that the given sides form a right triangle, we need to check the Pythagorean theorem.

Verification using the Pythagorean Theorem:

c^2  a^2   b^225^2  24^2   7^2625  576   49625  625

The equation holds true, confirming that triangle ABC is indeed a right triangle.

Step 2: Find the Angles

In a right triangle, one angle is 90°. Let's denote:

A as the angle opposite side a B as the angle opposite side b C as the right angle, i.e., C 90°

Finding Angle A

Using the sine function:

sinA a/c 24/25

To find A:

A arcsin(24/25)

Calculating A:

A ≈ 61.0°

Finding Angle B

Using the cosine function:

cosB b/c 7/25

To find B:

B arccos(7/25)

Calculating B:

B ≈ 28.0°

Step 3: Summary of the Results

Thus, the angles of triangle ABC are approximately:

A ≈ 61.0° B ≈ 28.0° C 90°

Final results:

Sides: a 24, b 7, c 25 Angles: A ≈ 61.0°, B ≈ 28.0°, C 90°

Since 7:24:25 is a basic Pythagorean triplet, with 7^2 24^2 25^2, and c is the hypotenuse with a right angle at vertex C.

Angle Calculations:

A arcsin(24/25) ≈ 73.74°

The other angle can be found using complementary angles, as the angles in a triangle sum to 180°. Alternatively, using a calculator:

B 180° - 73.74° 16.26°

This solution gives us a comprehensive look at solving the right triangle using both trigonometric functions and the Pythagorean theorem.