Solve each right triangle. In each case, C = 90°. If angle information is given in degrees and minutes, give answers in the same way. If angle information is given in decimal degrees, do likewise in answers. When two sides are given, give angles in degrees and minutes. See Examples 1 and 2.a = 958 m, b = 489 m

Verified step by step guidance

1. Identify the given sides of the right triangle: side \( a = 958 \) m and side \( b = 489 \) m.
2. Use the Pythagorean theorem to find the hypotenuse \( c \): \( c = \sqrt{a^2 + b^2} \).
3. Calculate angle \( A \) using the tangent function: \( \tan(A) = \frac{a}{b} \).
4. Find angle \( A \) by taking the arctangent: \( A = \tan^{-1}\left(\frac{a}{b}\right) \).
5. Determine angle \( B \) using the fact that the sum of angles in a triangle is 180°: \( B = 90° - A \).

Key Concepts

Right Triangle Properties

A right triangle is defined by one angle measuring 90 degrees. The other two angles are acute and their sum is always 90 degrees. The sides of a right triangle are categorized as the opposite, adjacent, and hypotenuse, with the hypotenuse being the longest side opposite the right angle. Understanding these properties is essential for applying trigonometric ratios.

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), defined as sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. These ratios are crucial for finding unknown angles and sides in right triangles.

Angle Measurement

Angles can be measured in degrees or radians, with degrees often expressed in degrees and minutes for precision. In this context, angles are given in degrees and minutes, which requires converting decimal degrees to this format when necessary. Understanding how to convert and express angles correctly is vital for accurate calculations in trigonometry.