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 = 51.7°, a = 28.1 ft

Verified step by step guidance

1. Identify the given elements of the right triangle: angle \( B = 51.7^\circ \), side \( a = 28.1 \) ft, and the right angle \( C = 90^\circ \). Recall that side \( a \) is opposite angle \( A \), side \( b \) is opposite angle \( B \), and side \( c \) is the hypotenuse opposite the right angle \( C \).
2. Calculate angle \( A \) using the fact that the sum of angles in a triangle is \( 180^\circ \). Since \( C = 90^\circ \), use the formula: \( A = 90^\circ - B \).
3. Use the Law of Sines to find the hypotenuse \( c \). The Law of Sines states: \( \frac{a}{\sin A} = \frac{c}{\sin C} \). Since \( C = 90^\circ \), \( \sin C = 1 \), so \( c = \frac{a}{\sin A} \).
4. Find side \( b \) using the Law of Sines again or by using the Pythagorean theorem. Using Law of Sines: \( b = c \times \sin B \). Alternatively, use \( b = \sqrt{c^2 - a^2} \).
5. Express all answers with the correct units and in decimal degrees as given. Since the angle \( B \) was given in decimal degrees, keep angles \( A \) and \( B \) in decimal degrees, and sides in feet.

Key Concepts

Right Triangle Properties

A right triangle has one angle equal to 90°, and the other two angles sum to 90°. Knowing one acute angle and a side allows the use of trigonometric ratios to find unknown sides and angles. The right angle simplifies calculations and ensures the Pythagorean theorem applies.

Trigonometric Ratios (Sine, Cosine, Tangent)

Sine, cosine, and tangent relate the angles of a right triangle to the ratios of its sides. For example, sine of an angle equals the opposite side over the hypotenuse. These ratios help find missing sides or angles when some measurements are known.

Angle Measurement and Conversion

Angles can be expressed in degrees and minutes or decimal degrees. Understanding how to convert between these formats is essential for accurate answers. For instance, 0.7° equals 42 minutes (0.7 × 60), ensuring consistency with the problem's requirements.