In Exercises 49–54, find the measure of the side of the right triangle whose length is designated by a lowercase letter. Round answers to the nearest whole number.

Verified step by step guidance

1. Identify the sides of the triangle: PR is the adjacent side, PQ is the hypotenuse, and QR is the opposite side to the angle at P.
2. Since we need to find the length of side QR, which is opposite to the given angle, use the tangent function: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
3. Substitute the given values into the tangent formula: \( \tan(21^\circ) = \frac{QR}{413} \).
4. Solve for QR by multiplying both sides by 413: \( QR = 413 \times \tan(21^\circ) \).
5. Use a calculator to find \( \tan(21^\circ) \) and multiply by 413 to find the length of QR, rounding to the nearest whole number.

Key Concepts

Right Triangle Properties

A right triangle has one angle measuring 90 degrees, and the other two angles are complementary, summing to 90 degrees. The sides of a right triangle are categorized as the hypotenuse (the side opposite the right angle) and the two legs (the other two sides). Understanding these properties is essential for applying trigonometric functions to solve for unknown side lengths or 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). For a given angle in a right triangle, sine is the ratio of the opposite side to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side. These ratios are crucial for solving for unknown sides or angles.

Rounding and Precision

Rounding is the process of adjusting a number to a specified degree of accuracy, often to the nearest whole number. In the context of trigonometry, it is important to round answers appropriately to ensure clarity and precision in communication. Understanding how to round correctly is essential when presenting final answers, especially in practical applications such as construction or navigation.