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Chapter 5: Problem 13

Use the given values to find the values (if possible) of all six trigonometric functions. $$\tan \theta \text { is undefined, } \quad \sin \theta>0$$

Short Answer

Expert verified
The values of the six trigonometric functions are: \(\sin \theta = 1\), \(\cos \theta = 0\), \(\tan \theta\) is undefined, \(\csc \theta = 1\), \(\sec \theta\) is undefined, and \(\cot \theta\) is undefined.

Step by step solution

01

Analyze Given Information

From the information \(\tan \theta\) is undefined, and \(\sin \theta > 0\), we can infer that the angle \(\theta\) is \(\frac{\pi}{2}\) because the tangent function is undefined at \(\frac{\pi}{2}\) and the sine of any angle in the first quadrant, where \(\theta = \frac{\pi}{2}\) lies, is positive.
02

Find the Six Trigonometric Functions

Now, evaluate the six trigonometric functions for \(\theta = \frac{\pi}{2}\).\n1. \(\sin \theta = 1\) as sine of \(\frac{\pi}{2}\) is 1.\n2. \(\cos \theta = 0\) as cosine of \(\frac{\pi}{2}\) is 0.\n3. \(\tan \theta\) is undefined as per the problem statement and mathematics rules.\n4. \(\csc \theta = 1/\sin \theta = 1\) as cosecant is the reciprocal of the sine function.\n5. \(\sec \theta\) is undefined as secant is the reciprocal of the cosine function and in this case \(\cos \theta = 0\), so \(\sec \theta\) is undefined.\n6. \(\cot \theta\) is undefined because cotangent is the reciprocal of the tangent function and in this case \(\tan \theta\) is undefined, so \(\cot \theta\) is undefined.

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