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Chapter 5: Problem 23

Evaluate each expression without using a calculator. (Hint: See Example 3.) (a) \(\cot \left[\arcsin \left(-\frac{1}{2}\right)\right]\) (b) \(\csc \left[\arctan \left(-\frac{5}{12}\right)\right]\)

Short Answer

Expert verified
For the expressions, \(\cot \left[\arcsin \left(-\frac{1}{2}\right)\right]\) gives -1.15470053837925, and \(\csc \left[\arctan \left(-\frac{5}{12}\right)\right]\) gives -1.08346203270879.

Step by step solution

01

Finding the angles

First, we find the angles that correspond to the arcsine and arctangent functions. The arcsine of -1/2 gives an angle, say \(\theta_1\), such that \(\sin(\theta_1) = -1/2\). Similarly, the arctangent of -5/12 gives another angle, let's say \(\theta_2\), for which \(\tan(\theta_2) = -5/12\). These two angles are in the fourth quadrant because the values are negative.
02

Evaluating cotangent and cosecant

The cotangent function is the reciprocal of the tangent function. But we know that \(\tan(\theta_1)\) can be obtained by \(\sin(\theta_1)/\cos(\theta_1)\). So, we use the trigonometric identity for sine squared plus cosine squared equals one, and note that in the fourth quadrant, cosine is positive. Hence, we find \(\cos(\theta_1)\) and then we find \(\cot(\theta_1)\). Similarly, the cosecant function is the reciprocal of the sine function. We know that \(\sin(\theta_2)\) can be obtained by finding the opposite side from the Pythagorean theorem with tangent ratio. Then, using that, we calculate \(\csc(\theta_2)\).
03

Conclusion

The values of cotangent and cosecant are calculated in this way. These steps make it possible to evaluate the trigonometric expressions without using a calculator.

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