Chapter 6, Problem 1MC

To determine

Whether the statement “If $\sin \alpha >0$ and $\cot \alpha >0$, then $\alpha$ is in the first quadrant” is true or false.

Answer to Problem 1MC

The given statement is $\underset{\_}{\text{true}}$.

Explanation of Solution

The given statement is “If $\sin \alpha >0$ and $\cot \alpha >0$, then $\alpha$ is in the first quadrant”.

Suppose that $P\left(x,y\right)$ is a point other than the vertex on the terminal side of an angle $\alpha$.

The radius r is the distance between the origin and P.

All the trigonometric functions depend on quadrant where the terminal side of the angle lies. The values of a trigonometric function can be positive, negative or zero accordingly.

That is, the sign of the function depends only on the coordinates of P on terminal side of the angle.

In first quadrant, the length of the radius is always positive. That is, $r>0$.

Both the coordinates are positive in first quadrant. That is, $x,y>0$.

That is, in first quadrant, $x,y\text{ and }r>0$.

Since $\sin \alpha =\frac{y}{r}$ and $\cot \alpha =\frac{x}{y}$, the sign of the function values will be positive in first quadrant.

That is, if $\sin \alpha >0$ and $\cot \alpha >0$, then $\alpha$ is in the first quadrant.

Thus, the given statement is true.