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 = 39°09', c = 0.6231 m

Verified step by step guidance

1. Identify the given elements of the right triangle: angle \( B = 39^\circ 09' \) and side \( c = 0.6231 \) meters, where \( c \) is the hypotenuse since 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 \( A = 90^\circ - B \). Remember to subtract the minutes properly when working with degrees and minutes.
3. Apply the sine and cosine definitions to find the other two sides: \( a = c \times \sin(B) \) and \( b = c \times \cos(B) \). Convert angle \( B \) from degrees and minutes to decimal degrees if necessary for calculation, then convert back to degrees and minutes for the final answer if required.
4. Calculate side \( a \) using \( a = c \times \sin(B) \) and side \( b \) using \( b = c \times \cos(B) \), keeping units consistent.
5. Summarize the solution by listing all sides \( a, b, c \) and angles \( A, B, C \) with angles expressed in degrees and minutes as requested.

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 by providing a fixed reference.

Trigonometric Ratios and Functions

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 solve for unknown sides or angles when partial information is given.

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 calculations and final answers, especially when the problem specifies a particular format for reporting results.