Question

Match the following range of angles in which the given trigonometric ratios are positive.

• A. $\sin \theta ,\cos ec\theta$
• B. $\cos \theta ,sec\theta$
• C. $\tan \theta ,cot\theta$
• D. All of them

Answer 1

$\theta$ varies from $0$ to $2\pi$ in I, II, III and IV quadrant.

In first quadrant $(0,\frac{\pi }{2})$


Both $x$ and $y$ intercepts are positive, i.e., $a>0,b>0$

By definition of $\sin \theta ,\cos \theta ,\tan \theta ,\cot \theta ,\sec \theta$, and $cosec\theta$, all the trigonometric function are positive in first quadrant.

In second quadrant, $(\frac{\pi }{2},\pi )$

$x$ intercept in second quadrant is negative. Therefore, $a<0,b>0$

Hence, functions which includes the values of x - intercept or adjacent side are negative and all the other functions are positive.

By definition, we know that except $\sin \theta$ and $cosec\theta$, all the other functions use adjacent side in their trigonometric ratios. So, only $\sin \theta$ and $cosec\theta$ are positive in second quadrant.

In third quadrant $(\pi ,\frac{3\pi }{2})$


Both $x$ and $y$ - intercepts are negative. Therefore $a<0,b<0$

Both adjacent and opposite sides are negative.

So, trigonometric ratios which use both adjacent and opposite sides are positive.

Only $\tan \theta$ and $\cot \theta$ use both adjacent and opposite sides. Hence, $\tan \theta$ and $\cot \theta$ are positive in third quadrant.

In fourth quadrant $(\frac{3\pi }{2},2\pi )$

$x$ intercept is positive while $y$ intercepts negative. Therefore $a>0,b<0$

Functions which use the values of y- intercept gives negative values, remaining functions will give positive value. Hence, $\cos \theta$ and $\sec \theta$ are positive in fourth quadrant.