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Chapter 4: Problem 51

Use the properties of inverse trigonometric functions to evaluate the expression. $$\arcsin (\sin 3 \pi)$$

Short Answer

Expert verified
The value of \(\arcsin (\sin 3 \pi)\) is 0

Step by step solution

01

Understand the Inverse Trigonometric Function

The function given is the arcsine or inverse sine function. The inverse sine function, generally written as \(\arcsin(x)\) or \(\sin^{-1}(x)\), is the reverse operation of the sine function. This means it takes a value 'y' and gives the angle whose sine is 'y'. The output of \(\arcsin(x)\) is an angle in the interval \([- \frac{\pi}{2}, \frac{\pi}{2}]\).
02

Evaluate the Sine Function

First, evaluate the function within the arcsine, which is \(\sin(3\pi)\). The sine function has a period of \(2\pi\), which means that \(\sin(\theta + 2 \pi n) = \sin(\theta)\), where n is an integer. Hence, \(\sin(3\pi)\) is same as \(\sin(\pi)\), because \(3\pi = \pi + 2\pi\), and \(\sin(\pi) = 0\). So, \(\sin(3\pi) = 0\).
03

Evaluate the Arcsine Function

Now, we can replace \(\sin(3\pi)\) with the value we calculated, which is 0. So, the expression \(\arcsin(\sin(3 \pi))\) becomes \(\arcsin(0)\). The arcsine of 0 is the angle whose sine is 0. From our understanding of sine values in the unit circle, we know that \(\sin(0) = 0\). Therefore, \(\arcsin(0) = 0\).

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Most popular questions from this chapter

Determine whether the statement is true or false. Justify your answer. To find the reference angle for an angle \(\theta\) (given in degrees), find the integer \(n\) such that \(0 \leq 360^{\circ} n-\theta \leq 360^{\circ} .\) The difference \(360^{\circ} n-\theta\) is the reference angle.

Sketch a graph of the function and compare the graph of \(g\) with the graph of \(f(x)=\arcsin x\). $$g(x)=\arcsin (x-1)$$

For the simple harmonic motion described by the trigonometric function, find (a) the maximum displacement, (b) the frequency, (c) the value of \(d\) when \(t=5,\) and (d) the least positive value of \(t\) for which \(d=0 .\) Use a graphing utility to verify your results. $$d=9 \cos \frac{6 \pi}{5} t$$

Determine whether the statement is true or false. Justify your answer. The Leaning Tower of Pisa is not vertical, but when you know the angle of elevation \(\theta\) to the top of the tower as you stand \(d\) feet away from it, you can find its height \(h\) using the formula \(h=d \tan \theta\)

Fill in the blank. If not possible, state the reason. $$\text { As } x \rightarrow \infty, \text { the value of } \arctan x \rightarrow\text { _____ } .$$
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