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 = 30.4, c = 50.2

Verified step by step guidance

1. Identify the given values: a = 30.4 (one leg), c = 50.2 (hypotenuse).
2. Use the Pythagorean theorem to find the other leg: b^2 = c^2 - a^2.
3. Calculate the angle opposite to side a using sine: $\sin (A)=\frac{a}{c}$.
4. Find angle A using the inverse sine function: $A={\sin }^{-1}(\frac{a}{c})$.
5. Calculate the remaining angle B using the fact that the sum of angles in a triangle is 180 degrees: $B=90-A$.

Key Concepts

Right Triangle Properties

A right triangle has one angle measuring 90 degrees, and the sides are categorized as the opposite, adjacent, and hypotenuse. The hypotenuse is the longest side, opposite the right angle, while the other two sides are perpendicular to each other. Understanding these properties is essential for applying trigonometric functions to solve for unknown lengths and angles.

Trigonometric Ratios

Trigonometric ratios relate the angles of a triangle to the lengths of its sides. The primary ratios are sine (sin), cosine (cos), and tangent (tan), defined as sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. These ratios are fundamental for solving right triangles, allowing us to find unknown angles and side lengths.

Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). This can be expressed as a² + b² = c². This theorem is crucial for finding the lengths of sides when two sides are known, and it complements the use of trigonometric ratios in solving right triangles.