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Chapter 4: Problem 95

In Exercises \(91-96,\) find two solutions of the equation. Give your answers in degrees \(\left(0^{\circ} \leq \theta<360^{\circ}\right)\) and in radians \((0 \leq \theta<2 \pi) .\) Do not use a calculator. (a) \(\tan \theta=1 \qquad\) (b) \(\cot \theta=-\sqrt{3}\)

Short Answer

Expert verified
(a) \(theta = 45^\circ\), \(\frac{\pi}{4}\) radians, \(theta = 225^\circ\), \(\frac{5\pi}{4}\) radians. (b) \(theta = 120^\circ\), \(\frac{2\pi}{3}\) radians, \(theta = 300^\circ\), \(\frac{5\pi}{3}\) radians.

Step by step solution

01

Solve the first equation

For (a), given \(tan\ \theta =\ 1\), locate the angle where the tangent function equals to one. This occurs at \(\frac{\pi}{4}\) or \(45^\circ\) and \(\frac{5\pi}{4}\) or \(225^\circ\) in the interval \(0^\circ \leq \theta <\ 360^\circ\) or \(0 \leq \theta <\ 2\pi\).
02

Solve the second equation

For (b), given \(cot\ \theta = -\sqrt{3}\), locate the angle where the cotangent function equals to -\(\sqrt{3}\). This occurs at \(2\pi/3\) radians or \(120^\circ\) and \(5\pi/3\) or \(300^\circ\) in the interval \(0^\circ \leq \theta <\ 360^\circ\) or \(0 \leq \theta <\ 2\pi\).
03

Conversion between degrees and radians

To convert radians to degrees, we multiply by \((180/\pi)\), and to convert from degrees to radians, we multiply by \((\pi/180)\). When we apply this to our solutions, we discover that we already have them in the correct intervals and formats, so we don't need to adjust anything.

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Most popular questions from this chapter

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