Solve each problem. See Examples 1–4. Distance across a Lake To find the distance RS across a lake, a surveyor lays off length RT = 53.1 m, so that angle T = 32°10' and angle S = 57°50'. Find length RS.
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Verified step by step guidance

1. Identify the triangle involved: points R, S, and T form a triangle where RT is known, and angles at T and S are given. The goal is to find the length RS.
2. Convert the given angles from degrees and minutes to decimal degrees for easier calculation: 32°10' becomes 32 + 10/60 degrees, and 57°50' becomes 57 + 50/60 degrees.
3. Calculate the third angle at point R using the triangle angle sum property: \(\angle R = 180^\circ - \angle T - \angle S\).
4. Use the Law of Sines to relate the sides and angles in the triangle: \(\frac{RS}{\sin(\angle T)} = \frac{RT}{\sin(\angle R)}\).
5. Rearrange the Law of Sines formula to solve for RS: \(RS = \frac{RT \times \sin(\angle T)}{\sin(\angle R)}\). Substitute the known values and compute the sine values to find RS.

Key Concepts

Triangle Angle Sum Property

The sum of the interior angles in any triangle is always 180°. Knowing two angles allows you to find the third angle, which is essential for applying trigonometric laws to solve for unknown sides.

Law of Sines

The Law of Sines relates the lengths of sides of a triangle to the sines of their opposite angles. It states that (side/sin opposite angle) is constant for all sides, enabling calculation of unknown sides or angles when some sides and angles are known.

Angle Measurement and Conversion

Angles given in degrees and minutes must be accurately interpreted or converted to decimal degrees for calculation. Understanding how to handle these measurements ensures precise use of trigonometric functions in solving the problem.