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 \(B = 73.0^\circ\), side \(b = 128\) inches, and the right angle \(C = 90^\circ\).
2. Use the fact that the sum of angles in a triangle is \(180^\circ\). Since \(C = 90^\circ\), find angle \(A\) by calculating \(A = 180^\circ - 90^\circ - B\).
3. Apply the Law of Sines to find side \(a\): \(\frac{a}{\sin B} = \frac{b}{\sin A}\). Rearrange to solve for \(a\): \(a = b \times \frac{\sin B}{\sin A}\).
4. Find the hypotenuse \(c\) using the Pythagorean theorem: \(c = \sqrt{a^2 + b^2}\).
5. Express all angles in degrees and minutes if needed, converting decimal degrees by multiplying the decimal part by 60 to get minutes.

Key Concepts

Right Triangle Properties

A right triangle has one angle exactly 90°, which simplifies calculations since the other two angles sum to 90°. Knowing one acute angle and a side allows use of trigonometric ratios to find unknown sides and angles. The right angle also enables the use of the Pythagorean theorem to relate side lengths.

Trigonometric Ratios (Sine, Cosine, Tangent)

Trigonometric ratios 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 allow calculation of unknown sides or angles when at least one side and one angle are known, essential for solving the triangle.

Angle Measurement and Conversion

Angles can be expressed in degrees and minutes or decimal degrees. Understanding how to convert between these formats is crucial for accurate answers. For instance, 1 degree equals 60 minutes, so converting decimal degrees to degrees and minutes involves separating the integer part and converting the decimal fraction accordingly.