Solve each right triangle. In Exercise 46, give angles to the nearest minute. In Exercises 47 and 48, label the triangle ABC as in Exercises 45 and 46. A = 39.72°, b = 38.97 m

Verified step by step guidance

1. Identify the given elements of the right triangle: angle \(A = 39.72^\circ\) and side \(b = 38.97\) m. Since this is a right triangle, one angle is \(90^\circ\), and the other two angles sum to \(90^\circ\).
2. Recall the labeling convention: side \(a\) is opposite angle \(A\), side \(b\) is opposite angle \(B\), and side \(c\) is the hypotenuse opposite the right angle \(C = 90^\circ\).
3. Use the sine function to find the hypotenuse \(c\) because \(\sin A = \frac{a}{c}\) and side \(b\) is given, so first find angle \(B = 90^\circ - A\) to relate side \(b\) to \(c\) using \(\sin B = \frac{b}{c}\).
4. Calculate the length of the hypotenuse \(c\) using \(c = \frac{b}{\sin B}\), where \(B = 90^\circ - 39.72^\circ\).
5. Find side \(a\) using the Pythagorean theorem \(a = \sqrt{c^2 - b^2}\) or use the sine function \(a = c \sin A\). Finally, express all angles to the nearest minute by converting decimal degrees to degrees and minutes.

Key Concepts

Right Triangle Properties

A right triangle has one angle of 90°, and the other two angles sum to 90°. Understanding this helps in determining unknown angles or sides when some parts are given, using the fact that the sum of all angles is 180°.

Trigonometric Ratios (Sine, Cosine, Tangent)

Trigonometric ratios relate the angles of a right triangle to the lengths of its sides. For example, sine of an angle is opposite side over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. These ratios are essential for finding missing sides or angles.

Angle Measurement and Conversion

Angles can be measured in degrees, minutes, and seconds, where 1 degree = 60 minutes. Converting decimal degrees to degrees and minutes is important for precise answers, especially when the problem asks for angles to the nearest minute.