In Exercises 30–32, 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.
2. Use the tangent function to find QR: \( \tan(48^\circ) = \frac{QR}{PR} \).
3. Substitute the known value of PR: \( \tan(48^\circ) = \frac{QR}{700} \).
4. Solve for QR: \( QR = 700 \times \tan(48^\circ) \).
5. Calculate \( QR \) 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 opposite side, adjacent side, and hypotenuse, which is the longest side opposite the right angle. Understanding these properties is essential for applying trigonometric functions to solve for unknown lengths.

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 example, in a right triangle, sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. These ratios are crucial for calculating unknown side lengths when given an angle and another side length.

Angle of Elevation

The angle of elevation is the angle formed by a horizontal line and the line of sight to an object above the horizontal line. In the context of right triangles, this angle helps determine the height of an object when the distance from the object is known. In this problem, the angle of 48° at point P is used to find the height of side QR using trigonometric functions.