Question

\(\triangle \mathrm{ABC}\) has a right angle at \(\mathrm{A}, B=25^{\circ}\) and \(\mathrm{AC}=17.2 \mathrm{~cm} .\) Solve \(\triangle \mathrm{ABC}\).

Short answer

Answer: In the right triangle ABC, the length of side AB is approximately 7.31 cm, the length of side BC is approximately 15.44 cm, and the measure of angle C is 65°.

Step-by-step solution

Find side AB using sine function

To find side AB, we use the sine function with the given angle B and the given side AC. The formula is as follows: AB = AC * sin(B) Plugging in our given values, we get: AB = 17.2 * sin(25°) Approximately, AB = 7.31 cm

Find side BC using cosine function

To find side BC, we use the cosine function with the same angle B and side AC. The formula reads: BC = AC * cos(B) Now, we plug in our values: BC = 17.2 * cos(25°) Approximately, BC = 15.44 cm

Find angle C

Since the sum of angles in a triangle adds up to 180 degrees and we know that angle A is a right angle (90°), and angle B is 25°, we can find angle C by subtracting these angles from 180°. C = 180° - A - B C = 180° - 90° - 25° C = 65° We have now solved the triangle. The right triangle ABC has sides AB ≈ 7.31 cm, BC ≈ 15.44 cm and angle C is 65°.

Key concepts

Right Triangle

A right triangle is a type of triangle that features a 90-degree angle. In the context of triangle ABC, angle A is 90 degrees, confirming that triangle ABC is indeed a right triangle. The right angle is crucial because it allows us to apply specific trigonometric functions that are tailored for right triangles.

In any right triangle:

• One angle is always 90 degrees.
• The side opposite the right angle is called the hypotenuse, which is always the longest side. In our triangle ABC, side AC is the hypotenuse.
• The other two sides are known as the legs of the triangle.
Understanding these basic properties is essential for solving right triangles utilizing trigonometric approaches.

Triangle Solving

Solving a triangle involves finding the unknown sides and angles. For right triangle ABC, we wanted to determine sides AB and BC as well as angle C. This process provides a comprehensive understanding of the triangle's dimensions and angles.

In our example:

• We knew angle B was 25 degrees, and we used trigonometric functions to find the lengths of the unknown sides.
• To solve for the unknown angle C, we leveraged the fundamental property in triangles: the sum of all angles in a triangle adds up to 180 degrees.

Through these steps, we effectively uncovered all components of triangle ABC.

Sine and Cosine Functions

The sine and cosine functions are the core trigonometric tools for solving right triangles. They relate the angles of a triangle to its side lengths, offering a simple way to compute unknown measurements.

For angle B in triangle ABC:

• The sine function is defined as the ratio of the opposite side to the hypotenuse. In our example, we used: \( \sin(B) = \frac{\text{AB}}{\text{AC}} \), allowing us to compute side AB.
• The cosine function involves the ratio of the adjacent side to the hypotenuse, expressed as: \( \cos(B) = \frac{\text{BC}}{\text{AC}} \), which helped us find side BC.

These two functions are foundational in trigonometry, providing powerful methods to analyze and solve right triangles with known angles.