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.b = 32 ft, c = 51 ft

Verified step by step guidance

1. insert step 1: Identify the given information. We have a right triangle with angle C = 90°, side b = 32 ft (adjacent to angle A), and hypotenuse c = 51 ft.
2. insert step 2: Use the Pythagorean theorem to find the missing side a. The formula is a^2 + b^2 = c^2.
3. insert step 3: Substitute the known values into the Pythagorean theorem: a^2 + 32^2 = 51^2.
4. insert step 4: Solve for a by rearranging the equation: a^2 = 51^2 - 32^2, and then take the square root of both sides to find a.
5. insert step 5: Use trigonometric ratios to find the angles. For angle A, use \( \cos A = \frac{b}{c} \) and for angle B, use \( \sin B = \frac{b}{c} \). Convert the angles to degrees and minutes if necessary.

Key Concepts

Right Triangle Properties

A right triangle has one angle measuring 90 degrees, and the other two angles must sum to 90 degrees. The sides of a right triangle are referred to as the opposite, adjacent, and hypotenuse. The hypotenuse is the longest side, opposite the right angle, while the other two sides are used to define the angles. Understanding these properties is essential for solving for unknown angles and sides.

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 calculating unknown angles and sides in right triangles, especially when two sides are known.

Angle Measurement Conversions

Angles can be measured in degrees and minutes or in decimal degrees. One degree is divided into 60 minutes, where 1 minute equals 1/60 of a degree. When solving triangles, it is important to maintain consistency in angle measurement formats. Converting between these formats may be necessary to provide answers in the required form, ensuring clarity and accuracy in the results.