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Chapter 7: Problem 18

Find the following exactly in radians and degrees. $$\sin ^{-1} \frac{\sqrt{3}}{2}$$

Short Answer

Expert verified
The answer is \( \frac{\text{pi}}{3} \) radians or 60 degrees.

Step by step solution

01

Understand the Inverse Function

The inverse sine function, denoted \(\frac{\text{sin}^{-1}}{x}\), tells us the angle whose sine is \(x\). We need to find an angle \( \theta \) such that \( \text{sin}(\theta) = \frac{\text{sqrt(3)}}{2} \).
02

Recall Key Sine Values

Recall that \( \text{sin}( \frac{ \text{pi} }{3} ) = \frac{\text{sqrt(3)}}{2} \). This means that \( \theta \) must be \( \frac{\text{pi}}{3} \) radians.
03

Convert Radians to Degrees

To convert radians into degrees, recall the conversion factor \( 1 \text{ radian} = \frac{180}{\text{pi}} \text{ degrees} \). Thus, \( \frac{\text{pi}}{3} \) radians can be converted to degrees as follows: \( \frac{\text{pi}}{3} \times \frac{180}{\text{pi}} = 60 \text{ degrees} \).
04

Verify the Result

We verify that the angle \( \frac{\text{pi}}{3} \) radians or 60 degrees correctly satisfies the given inverse sine condition since \( \text{sin}(60^\text{o}) = \frac{\text{sqrt(3)}}{2} \).

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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, also called arcsine and denoted as \(\sin^{-1}(x)\), helps us find the angle when the sine value is known. This means if we have \(\sin(\theta) = x\), then \(\theta = \sin^{-1}(x)\).
In our exercise, we need an angle \(\theta\) such that \(\sin(\theta) = \frac{\sqrt{3}}{2}\). The sine of which angle equals \(\frac{\sqrt{3}}{2}\)?
The answer lies within the principal range of the inverse sine function, which is \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) radians. We'll explore how to match the given value to a known angle in this range.
key sine values
To solve the problem using the inverse sine function, recall some key sine values. By knowing these standard values, you can quickly identify angles without using a calculator.
Here are some important ones to remember:
  • \sin(0) = 0\
  • \sin(\frac{\pi}{6}) = \frac{1}{2}\
  • \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\
  • \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}\
  • \sin(\frac{\pi}{2}) = 1\
    • For our specific exercise, matched to \(\sin\left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2}\). Hence, \(\theta = \frac{\pi}{3}\) radians.
    radian to degree conversion
    Now, let's convert \(\frac{\pi}{3}\) radians to degrees.
    Remember, the radian is the standard unit of angular measure used in many areas of mathematics. The conversion factor between radians and degrees is \( 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \).
    To convert \(\frac{\pi}{3}\) radians to degrees, use:
    \frac{\pi}{3} \times \frac{180}{\pi} = 60 \text{ degrees}\
    This means our angle \(\frac{\pi}{3}\) radians can also be represented as 60 degrees.
    Thus, \(\sin^{-1}\left( \frac{\sqrt{3}}{2} \right) = 60^{\circ} \).

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    Most popular questions from this chapter

    Electrical Theory. In electrical theory, the following equations occur: $$E_{1}=\sqrt{2} E_{t} \cos \left(\theta+\frac{\pi}{P}\right)$$ and $$E_{2}=\sqrt{2} E_{t} \cos \left(\theta-\frac{\pi}{P}\right)$$. Assuming that these equations hold, show that $$\frac{E_{1}+E_{2}}{2}=\sqrt{2} E_{t} \cos \theta \cos \frac{\pi}{P}$$ and $$\frac{E_{1}-E_{2}}{2}=-\sqrt{2} E_{t} \sin \theta \sin \frac{\pi}{P}$$.

    Evaluate. \(\cos \left(\sin ^{-1} \frac{\sqrt{2}}{2}+\cos ^{-1} \frac{3}{5}\right)\)

    Simplify. Check your results using a graphing calculator. $$\frac{\cos x-\sin \left(\frac{\pi}{2}-x\right) \sin x}{\cos x-\cos (\pi-x) \tan x}$$

    Solve, finding all solutions in \([0,2 \pi)\). $$5 \cos 2 x+\sin x=4$$,.$$

    Assuming that \(\sin u=\frac{3}{5}\) and \(\sin v=\frac{4}{5}\) and that \(u\) and \(v\) are between 0 and \(\pi / 2,\) evaluate each of the following exactly. $$\cos (u+v)$$
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