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 = 73.0°, b = 128 in.

Verified step by step guidance

1. Identify the given information: angle \( B = 73.0^\circ \) and side \( b = 128 \) inches.
2. Since \( C = 90^\circ \), use the fact that the sum of angles in a triangle is \( 180^\circ \) to find angle \( A \): \( A = 180^\circ - B - C \).
3. Use the sine function to find side \( a \): \( \sin(A) = \frac{a}{c} \), where \( c \) is the hypotenuse.
4. Use the cosine function to find side \( c \): \( \cos(B) = \frac{b}{c} \).
5. Use the Pythagorean theorem to verify the sides: \( a^2 + b^2 = c^2 \).

Key Concepts

Right Triangle Properties

In a right triangle, one angle measures 90 degrees, and the sum of the other two angles must equal 90 degrees. This property allows us to use the relationships between the angles and sides to find unknown values. The Pythagorean theorem also applies, stating that the square of the hypotenuse equals the sum of the squares of the other two sides.

Trigonometric Ratios

Trigonometric ratios relate the angles of a triangle to the lengths of its sides. The primary ratios are sine, cosine, and tangent, defined as the ratios of the lengths of the opposite side to the hypotenuse, adjacent side to the hypotenuse, and opposite side to the adjacent side, respectively. These ratios are essential for calculating unknown angles and sides in right triangles.

Angle Measurement Conversions

Angles can be measured in degrees and minutes or in decimal degrees. Understanding how to convert between these formats is crucial for solving problems accurately. For example, 1 degree equals 60 minutes, and when given angles in different formats, one must convert them to a consistent format to perform calculations correctly.