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Chapter 0: Problem 18

Find the exact value of the trigonometric functions at the indicated angle. \(\sin \theta, \cos \theta\), and \(\csc \theta\) for \(\theta=-\pi / 4\)

Short Answer

Expert verified
For \(\theta = -\frac{\pi}{4}\): \(\sin\theta = -\frac{\sqrt{2}}{2}\), \(\cos\theta = \frac{\sqrt{2}}{2}\), and \(\csc\theta = -\sqrt{2}\).

Step by step solution

01

Recall the properties of the unit circle and trigonometric functions

In the unit circle, the x-coordinate of a point corresponds to the cosine of the angle and the y-coordinate corresponds to the sine of the angle. Recall that for any angle \(\theta\), \(\sin\theta = y\), \(\cos\theta = x\), and \(\csc\theta = \frac{1}{\sin\theta}\). Since the given angle is \(-\frac{\pi}{4}\), we need to find the coordinates of the point on the unit circle at this angle.
02

Find the coordinates for the given angle

To find the coordinates, we make note of the fact that the angle \(-\frac{\pi}{4}\) is the negative of \(\frac{\pi}{4}\), which is a 45-degree angle. A 45-degree angle on the unit circle has coordinates \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\). Since the angle is negative, we move clockwise from the positive x-axis by 45 degrees starting from \((1, 0)\), which places us in the fourth quadrant. In the fourth quadrant, the x-coordinate is still positive and the y-coordinate is negative. Thus, the coordinates of the point \(\theta\) are \(\left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)\).
03

Calculate the values for the trigonometric functions

Now, we apply the unit circle definition for each trigonometric function. \(\sin\theta = y = -\frac{\sqrt{2}}{2}\) \(\cos\theta = x = \frac{\sqrt{2}}{2}\) \(\csc\theta = \frac{1}{\sin\theta} = -\frac{2}{\sqrt{2}} = -\sqrt{2}\), where we rationalize the denominator by multiplying both the numerator and the denominator by \(\sqrt{2}\).
04

Present the values of the trigonometric functions

Final answers for the trigonometric functions at \(\theta = -\frac{\pi}{4}\): \(\sin\theta = -\frac{\sqrt{2}}{2}\) \(\cos\theta = \frac{\sqrt{2}}{2}\) \(\csc\theta = -\sqrt{2}\)

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Most popular questions from this chapter

Let \(f(x)=2 x^{3}-5 x^{2}+x-2\) and \(g(x)=2 x^{3}\). a. Plot the graph of \(f\) and \(g\) using the same viewing window: \([-5,5] \times[-5,5]\). b. Plot the graph of \(f\) and \(g\) using the same viewing window: \([-50,50] \times[-100,000,100,000] .\) c. Explain why the graphs of \(f\) and \(g\) that you obtained in part (b) seem to coalesce as \(x\) increases or decreases without bound. Hint: Write \(f(x)=2 x^{3}\left(1-\frac{5}{2 x}+\frac{1}{2 x^{2}}-\frac{1}{x^{3}}\right)\) and study its behavior for large values of \(x\).

Find the exact value of the given expression. $$ \sin ^{-1} 0 $$

a. Plot the graph of \(f(x)=\cos (\sin x)\). Is \(f\) odd or even? b. Verify your answer to part (a) analytically.

Find the exact value of the given expression. $$ \sin \left(\sin ^{-1} \frac{1}{\sqrt{2}}\right) $$

Find the zero(s) of the function f to five decimal places. $$ f(x)=2 x^{4}-4 x^{2}+1 $$
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