Solve each right triangle. In Exercise 46, give angles to the nearest minute. In Exercises 47 and 48, label the triangle ABC as in Exercises 45 and 46.A = 39.72°, b = 38.97 m

Verified step by step guidance

1. Identify the given information: angle \( A = 39.72^\circ \) and side \( b = 38.97 \text{ m} \).
2. Since it's a right triangle, angle \( C = 90^\circ \). Use the fact that the sum of angles in a triangle is \( 180^\circ \) to find angle \( B \): \( B = 180^\circ - A - C \).
3. Use the sine function to find side \( a \): \( \sin(A) = \frac{a}{c} \), where \( c \) is the hypotenuse. Rearrange to find \( a = c \cdot \sin(A) \).
4. Use the cosine function to find side \( c \): \( \cos(A) = \frac{b}{c} \). Rearrange to find \( c = \frac{b}{\cos(A)} \).
5. Use the tangent function to find side \( a \): \( \tan(A) = \frac{a}{b} \). Rearrange to find \( a = b \cdot \tan(A) \).

Key Concepts

Right Triangle Properties

Right triangles have specific properties that make them unique, including the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Understanding these properties is essential for solving for unknown sides and angles in right triangles.

Trigonometric Ratios

Trigonometric ratios, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. For 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 fundamental for finding unknown angles and sides.

Angle Measurement

Angles in trigonometry can be measured in degrees or radians, with degrees being the more common unit in basic applications. In this context, the problem specifies giving angles to the nearest minute, which is a subdivision of degrees (1 degree = 60 minutes). Understanding how to convert and express angles accurately is crucial for precise calculations in triangle problems.