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 = 958 m, b = 489 m

Verified step by step guidance

1. Identify the given elements of the right triangle: side \(a = 958\) m, side \(b = 489\) m, and the right angle \(C = 90^\circ\).
2. Use the Pythagorean theorem to find the length of the hypotenuse \(c\): \(c = \sqrt{a^2 + b^2} = \sqrt{958^2 + 489^2}\).
3. Calculate angle \(A\) using the tangent function, since you know the opposite side \(a\) and adjacent side \(b\): \(\tan A = \frac{a}{b}\), so \(A = \arctan\left(\frac{958}{489}\right)\).
4. Convert the angle \(A\) from decimal degrees to degrees and minutes if necessary, by separating the integer part (degrees) and multiplying the decimal part by 60 to get minutes.
5. Find angle \(B\) by subtracting angle \(A\) from \(90^\circ\): \(B = 90^\circ - A\), and convert to degrees and minutes if needed.

Key Concepts

Right Triangle Properties

A right triangle has one angle equal to 90°, which simplifies calculations since the other two angles must sum to 90°. Knowing this helps in applying trigonometric ratios and the Pythagorean theorem to find missing sides or angles.

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 is opposite side over hypotenuse. These ratios allow calculation of unknown angles or sides 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 communication and calculation, especially when the problem specifies the required format for answers.