Chapter 4.2, Problem 92E

a.

To determine

If all the trigonometric function of $t$ exist and then to explain the reasoning.

Answer to Problem 92E

The answer is yes.

Explanation of Solution

Given Information: A relevant figure.

As $(x,y)$ neither lies on $x-axis$ nor on $y-axis$ , none of them equals $0$ ,

  $x\ne 0,y\ne 0$

All of the trigonometric functions exist for $t$ , the reason is that the denominator will not be $0$ for any trigonometric function.

Thus, the answer is yes.

b.

To determine

If the sign of the trigonometric functions will be positive or negative and then to explain the reasoning.

Answer to Problem 92E

Only tangent and cotangent will be positive.

Explanation of Solution

Given Information: A relevant figure and results from previous part.

As the terminal point lies in the third quadrant, only tangent and cotangent will be positive, all other trigonometric functions will be negative. This is so because in third quadrant both $x$ and $y$ are negative, in case of tangent and cotangent negative sign cancels out $\left(\tan (t)=\frac{y}{x}and\cot (t)=\frac{x}{y}\right)$ while for others it ultimately persists.