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: RP is the adjacent side, RQ is the opposite side, and PQ is the hypotenuse.
2. Use the tangent function, which relates the opposite side to the adjacent side: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
3. Substitute the known values into the tangent formula: \( \tan(39^\circ) = \frac{RQ}{51} \).
4. Solve for RQ by multiplying both sides by 51: \( RQ = 51 \times \tan(39^\circ) \).
5. Calculate the value of \( RQ \) using a calculator and round 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 find 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, sin is the ratio of the opposite side to the hypotenuse, cos is the ratio of the adjacent side to the hypotenuse, and tan is the ratio of the opposite side to the adjacent side. These ratios are crucial for solving for unknown sides or angles.

Rounding and Measurement

In practical applications, measurements often need to be rounded to a specified degree of accuracy. In this context, rounding to the nearest whole number means adjusting the calculated side length to the closest integer. This is important for ensuring that the final answer is practical and usable in real-world scenarios, especially in fields like construction or engineering.