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Chapter 4: Problem 45

Find the exact value of each trigonometric function. Do not use a calculator. $$\cot \left(-\frac{5 \pi}{4}\right)$$

Short Answer

Expert verified
The exact value of \( \cot(-5pi/4) \) is -1.

Step by step solution

01

Identify the Quadrant of the Angle

The given angle is -5pi/4. As the negative sign indicates, it's measured clockwise from the positive x-axis. Simply neglecting the negative sign, when we check 5pi/4, it is more than pi but less than 2pi, this indicates that the angle is in 3rd quadrant. Since it's a negative angle, we move clockwise direction to reach the 3rd quadrant.
02

Find the corresponding angle in the First Quadrant

The corresponding angle in the positive direction for -5pi/4 can be calculated as |(-5pi/4) mod (2pi)|, which is pi/4. Note that for any angle \( \theta \), \(\cot(-\theta) = -\cot(\theta)\) due to the property of the cotangent function.
03

Compute the cotangent of the angle

The cotangent of an angle \( \theta \) in quadrant 3 is negative, but we just found that our corresponding angle was \( \theta \) = pi/4. So, the cotangent will be determined as \( -\cot(pi/4) \). We know that \( \cot(pi/4) = 1 \), since the x and y coordinates of point on the unit circle for pi/4 are equal.
04

Write the final answer

Combining the information from the above steps, we get the final solution as \( -\cot(pi/4) = -1 \)

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Most popular questions from this chapter

Graph \(f, g,\) and \(h\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi .\) Obtain the graph of h by adding or subtracting the corresponding \(y\) -coordinates on the graphs of \(f\) and \(g\) $$f(x)=-2 \sin x, g(x)=\sin 2 x, h(x)=(f+g)(x)$$

Use a vertical shift to graph one period of the function. $$y=\cos x+3$$

will help you prepare for the material covered in the next section. $$\text { Solve: } \quad-\frac{\pi}{2}

Use a graphing utility to graph \(y=\sin x\) and \(y=x-\frac{x^{3}}{6}+\frac{x^{5}}{120}\) in a \(\left[-\pi, \pi, \frac{\pi}{2}\right]\) by [-2,2,1] viewing rectangle. How do the graphs compare?

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=-3 \cos \left(2 x-\frac{\pi}{2}\right)$$
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