/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 Find \(\alpha\) to the nearest t... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 25

Find \(\alpha\) to the nearest tenth of a degree, where \(-90^{\circ} \leq \alpha \leq 90^{\circ}\). $$ \sin \alpha=-1 / 3 $$

Short Answer

Expert verified
\alpha \approx -19.5^{\circ}

Step by step solution

01

Understand the Problem

We need to find the angle \( \alpha \) such that \( \sin(\alpha) = -1/3 \) within the range \( -90^{\circ} \leq \alpha \leq 90^{\circ} \).
02

Use the Inverse Sine Function

The inverse sine function, \( \sin^{-1}(x) \), can be used to find an angle \( \alpha \).
03

Determine the Angle

Calculate \( \alpha = \sin^{-1}(-1/3) \) using a calculator.
04

Round to the Nearest Tenth

Round the calculated angle to the nearest tenth of a degree.

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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 is a tool we use to find an angle when given a sine value. In trigonometry, this is written as \(\text{sin}^{-1}(x)\). It allows us to solve for the angle \(\ \alpha\) between \(-90^{\text{\circ}}\) and \(90^{\text{\circ}}\). The inverse sine function undoes the sine function. So, if \(\ \sin(\theta) = y\), then \(\ \sin^{-1}(y) = \theta\).
Angle Calculation
To find the angle \(\text{\alpha}\) for which \(\ \sin(\text{\alpha}) = -1/3\), follow these steps: First, use a calculator with an inverse sine function. Input the number \(\ -1/3\), and it will return an angle \(\text{\alpha}\). For this problem, our calculator gives \(\text{\alpha} = -19.5^{\text{\circ}}\). This value falls within the specified range of \(-90^{\text{\circ}}\) to \(90^{\text{\circ}}\).
Rounding to Nearest Tenth
The last step is rounding the angle \(\text{\alpha}\) to the nearest tenth of a degree. Rounding involves looking at the first digit after the decimal point. If this digit is 5 or higher, we round up. Otherwise, we round down. In this problem, \(-19.5\) already ends in a \('.5\), so it remains as \(-19.5\). If you calculated an angle like \(-19.53\), for example, you would round down to \(-19.5\).

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Most popular questions from this chapter

Find the exact value of each composition without using a calculator or table. $$ \cot ^{-1}(\cot (\pi / 6)) $$

Find the exact value of each composition without using a calculator or table. $$ \sin ^{-1}(\cos (2 \pi / 3)) $$

Solve each equation. Round approximate answers to the nearest tenth of a degree. $$ 3 \cos (\alpha)+\sqrt{12}=5 \cos (\alpha)+\sqrt{3} \text { for }-360^{\circ} \leq \alpha \leq 360^{\circ} $$

Find all real numbers in the interval \([0,2 \pi)\) that satisfy each equation. Round approximate answers to the nearest tenth. $$ \csc x-\sqrt{3}=\cot x $$

Solve each equation for the indicated variable. Solve \(q=3 \sin (\pi b-\pi)\) for \(b\) where \(\frac{1}{2} \leq b \leq \frac{3}{2}\).
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