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. A = 53°24', c = 387.1 ft

Verified step by step guidance

1. Identify the given elements of the right triangle: angle \(A = 53^\circ 24'\) and side \(c = 387.1\) ft, where \(c\) is the hypotenuse opposite the right angle \(C = 90^\circ\).
2. Calculate the remaining angle \(B\) using the fact that the sum of angles in a triangle is \(180^\circ\). Since \(C = 90^\circ\), use the formula: \(B = 90^\circ - A\). Remember to subtract the minutes properly.
3. Use the sine and cosine trigonometric ratios to find the lengths of the legs \(a\) and \(b\). Specifically, use \(a = c \times \sin(A)\) and \(b = c \times \cos(A)\), where \(a\) is opposite angle \(A\) and \(b\) is adjacent to angle \(A\).
4. Convert the decimal results of \(a\) and \(b\) into feet, keeping the units consistent, and round as appropriate based on the problem context.
5. Express the calculated angle \(B\) in degrees and minutes format, ensuring the minutes are correctly converted from any decimal part.

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 one side allows the use of trigonometric ratios to find unknown sides and angles. This foundational property guides the approach to solving the triangle.

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 enable calculation of unknown sides or angles when partial information is given, such as one angle and one side.

Angle Measurement in Degrees and Minutes

Angles can be expressed in degrees and minutes, where 1 degree equals 60 minutes. When solving triangles with angles given in degrees and minutes, it is important to maintain this format in answers. Conversions between decimal degrees and degrees-minutes format may be necessary for accuracy and consistency.