Problem 23 - Calculus of a Single Variable

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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 \(\csc(\theta_2)\).

Step 3: 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.