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Chapter 8: Problem 11

Find the measure of each angle to the nearest tenth of a degree. $$\sin B=0.6307$$

Short Answer

Expert verified
The angle measure is approximately 39.0 degrees.

Step by step solution

01

Understand the Problem

We need to find the measure of the angle \( B \) given \( \sin B = 0.6307 \). This means we are looking for an angle whose sine is 0.6307.
02

Use the Inverse Sine Function

The inverse sine function, denoted as \( \sin^{-1} \), allows us to find the angle when we know the sine value. We will use \( \sin^{-1}(0.6307) \) to find \( B \).
03

Calculate the Angle

Using a calculator, compute \( B = \sin^{-1}(0.6307) \). Make sure the calculator is in degree mode. This calculation yields \( B \approx 39.0 \) degrees.
04

Verify the Result

Check if the sine of the calculated angle is approximately 0.6307. Compute \( \sin(39.0^\circ) \approx 0.6307 \) to ensure accuracy.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inverse Sine Function
The inverse sine function, often written as \( \sin^{-1} \) or \( \arcsin \), is a powerful tool in trigonometry. It allows us to find the angle that corresponds to a given sine value. This is especially useful when you know the sine of an angle but not the angle itself.
  • **Notation**: The inverse sine of a number \( x \) is denoted as \( \sin^{-1}(x) \).
  • **Range**: The output of the \( \sin^{-1} \) function is usually an angle in the range of \(-90^\circ\) to \(90^\circ\) in degrees (or \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) in radians).
  • **Calculators**: Most scientific calculators can compute this directly if they have a specific button or function for the inverse sine (\( \sin^{-1} \))鈥攋ust ensure it is set to the correct mode for degrees or radians depending on your requirement.
Remember, when you perform \( \sin^{-1}(0.6307) \), you are asking: "What angle has a sine of 0.6307?" In our problem, the answer was found to be approximately \(39.0^\circ\).
Angle Measurement
Angle measurement is crucial when dealing with trigonometric functions. Angles can be measured in degrees or radians.
  • **Degrees**: This is the most common unit of measurement, especially in everyday applications. A full circle is \(360^\circ\).
  • **Radians**: A more mathematical approach, where a full circle is \(2\pi\) radians. One radian approximately equals \(57.296^\circ\).
When using the inverse sine function, make sure that your calculator is set to the appropriate mode. In this exercise, we used degrees. Therefore, in your calculator settings, select "degree mode" for accurate results. Failing to do so may result in incorrect angle values.
Trigonometric Calculation
Trigonometric calculations involve finding the sides and angles of triangles, using the relationships between angles and trigonometric ratios like sine, cosine, and tangent.
For the given problem, you need to find angle \( B \) using the sine value \(0.6307\). Here鈥檚 how you proceed:
  • Calculate: Use the inverse sine function on a calculator: \( B = \sin^{-1}(0.6307) \).
  • Verify: After finding \( B \) using this function, check by taking the sine of \(39.0^\circ\). You should approximate back to \(0.6307\).
Why is verification important? Because small errors can occur in calculations or settings, which could throw your results off. Always ensure the final result aligns with your initial data to maintain accuracy.

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