Question

Find the sign of the expression if the terminal point determined by $t$is in the given quadrant.

$\cos t\sec t,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}anyquadrant}$

Short answer

In any quadrant, $\mathbf{cos}\mathit{t}\mathbf{sect}\mathbf{=}\mathbf{1}$.

Step-by-step solution

Step 1. State the definition of the terminal point on the unit circle.

Starting at the point $(1,0)$, if $t\ge 0$, then $t$is the distance along the unit circle in a clockwise direction and if $t<0$, then $\left|t\right|$is the distance along the unit circle in a clockwise direction. Here, $t$is a real number and this distance generates a point $P\left(x,y\right)$ on the unit circle. The point $P\left(x,y\right)$ obtained in this way is called the terminal point determined by the real number $t$.

Step 2. Find the sign-in quadrant II. 

Determine the sign of the trigonometric function of $t$by noting the quadrant in which the terminal point lies. The terminal point of $t$lies in any quadrant, and in any quadrant and with being either both positive or both negative.

Step 3. Find the sign of the expression.

Since using the reciprocal identity,$\cos t=\frac{1}{\sec t}$ in any quadrant, therefore, the product of the two will be positive and equal to 1 in any quadrant.