Question

in exercises 13 - 16, state the sign (+ or -) of (a) sin t, (b) cos t, and (c) tan t for values of t in the interval given. 13. (0, \frac{\pi}{2}) 14. (\frac{\pi}{2}, \pi)

Explanation:

Step1: Recall the unit - circle definitions

In the unit - circle, for a point \((x,y)\) on the terminal side of an angle \(t\), \(\sin t=y\), \(\cos t = x\), and \(\tan t=\frac{y}{x}\).

Step2: Analyze the interval \((0,\frac{\pi}{2})\)

In the interval \((0,\frac{\pi}{2})\) (first quadrant), for any angle \(t\) in this interval:

• For \(\sin t\):

Since \(y>0\) (in the first quadrant), \(\sin t>0\).

• For \(\cos t\):

Since \(x > 0\) (in the first quadrant), \(\cos t>0\).

• For \(\tan t\):

Since \(\tan t=\frac{\sin t}{\cos t}\) and \(\sin t>0\), \(\cos t>0\), then \(\tan t=\frac{\text{positive}}{\text{positive}}>0\).

Answer:

(a) \(+\)
(b) \(+\)
(c) \(+\)