In Exercises 1–12, solve the right triangle shown in the figure. Round lengths to two decimal places and express angles to the nearest tenth of a degree.A = 52.6°, c = 54

Verified step by step guidance

1. Identify the given values: angle A = 52.6° and hypotenuse c = 54.
2. Use the sine function to find the length of side opposite angle A: \( \sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} \).
3. Calculate the opposite side: \( \text{opposite} = \sin(52.6°) \times 54 \).
4. Use the cosine function to find the length of the adjacent side: \( \cos(A) = \frac{\text{adjacent}}{\text{hypotenuse}} \).
5. Calculate the adjacent side: \( \text{adjacent} = \cos(52.6°) \times 54 \).

Key Concepts

Right Triangle Properties

A right triangle has one angle measuring 90 degrees. The sides opposite the angles are referred to as the opposite side, adjacent side, and hypotenuse. The relationships between these sides are fundamental in trigonometry, allowing for the application of the Pythagorean theorem and trigonometric ratios (sine, cosine, and tangent) to solve for unknown lengths and angles.

Trigonometric Ratios

Trigonometric ratios relate the angles of a triangle to the lengths of its sides. For a right triangle, sine (sin), cosine (cos), and tangent (tan) are defined as follows: sin(A) = opposite/hypotenuse, cos(A) = adjacent/hypotenuse, and tan(A) = opposite/adjacent. These ratios are essential for calculating unknown side lengths and angles when given certain triangle parameters.

Angle Measurement and Rounding

In trigonometry, angles are typically measured in degrees or radians. When solving problems, it is often necessary to round the results to a specified degree of precision. In this case, angles should be expressed to the nearest tenth of a degree, which involves understanding how to round decimal values correctly to ensure accuracy in the final answer.