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 = 10.8, b = 24.7

Verified step by step guidance

1. Identify the given sides of the right triangle: $a=10.8$ and $b=24.7$.
2. Use the Pythagorean theorem $c=\sqrt{a^2+b^2}$ to find the hypotenuse $c$.
3. Calculate angle $\mathrm{\angle }P$ using ${\tan }^{-1}(\frac{a}{b})$.
4. Calculate angle $\mathrm{\angle }Q$ using ${90}^{\circ }-\mathrm{\angle }P$.
5. Round the lengths to two decimal places and express angles to the nearest tenth of a degree.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (r) is equal to the sum of the squares of the lengths of the other two sides (p and q). This relationship is expressed as r² = p² + q², which is essential for calculating the length of any side when the lengths of the other two sides are known.

Trigonometric Ratios

Trigonometric ratios relate the angles of a triangle to the lengths of its sides. In a right triangle, the primary ratios are sine (sin), cosine (cos), and tangent (tan). For example, sin(Q) = opposite/hypotenuse, cos(Q) = adjacent/hypotenuse, and tan(Q) = opposite/adjacent. These ratios are crucial for finding unknown angles and side lengths.

Angle Measurement

In trigonometry, angles are typically measured in degrees or radians. For this problem, angles must be expressed to the nearest tenth of a degree. Understanding how to convert between degrees and radians, as well as how to use a calculator to find angles based on trigonometric ratios, is vital for solving right triangles.