In Exercises 41–42, find a to the nearest tenth.
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Verified step by step guidance

1. Identify the specific trigonometric problem or triangle given in the exercise to understand what 'a' represents (e.g., a side length or an angle).
2. Determine which trigonometric function or theorem applies based on the information provided (such as sine, cosine, tangent, or the Pythagorean theorem).
3. Set up the appropriate equation using the chosen trigonometric ratio. For example, if you know an angle and a side, use \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\), or \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\).
4. Solve the equation algebraically for 'a' by isolating it on one side of the equation.
5. Use a calculator to evaluate the trigonometric function and then round the result for 'a' to the nearest tenth as required.

Key Concepts

Understanding the Variable 'a' in Trigonometric Context

In trigonometry problems, the variable 'a' often represents a side length or an angle measure within a triangle. Identifying what 'a' stands for is crucial to applying the correct formulas or theorems, such as the Law of Sines or Cosines, to find its value accurately.

Application of Trigonometric Ratios and Laws

Solving for 'a' typically involves using trigonometric ratios (sine, cosine, tangent) or laws like the Law of Sines or Law of Cosines. These relationships connect angles and sides in triangles, enabling calculation of unknown values when given sufficient information.

Rounding and Precision in Final Answers

The instruction to find 'a' to the nearest tenth highlights the importance of rounding in trigonometric calculations. After computing the value, it must be rounded appropriately to ensure the answer is both accurate and presented in a standardized format.