Question

Some trigonometric ratios and the interval in which $\theta$ lies is given. Match the intervals with the ratios which are positive in those intervals.

$\theta$ gives positive values

p. (0, $\frac{\pi }{2}$) 1. Only sin $\theta$, cosec$\theta$

q. ($\frac{\pi }{2}$, $\pi$) 2. Only cos$\theta$, sec$\theta$

r. $(\pi ,\frac{3\pi }{2})$ 3. Only tan$\theta$, cot$\theta$

s. $(\frac{3\pi }{2},2\pi )$ 4. All sin$\theta$, cos$\theta$, tan$\theta$, cot$\theta$, sec$\theta$, cosec$\theta$

• A. p - 1, q - 2, r - 3, s - 4
• B. p - 4, q - 3, r - 2, s - 1
• C. p - 4, q - 1, r - 2, s - 3
• D. p - 4, q - 1, r - 3, s - 2

Answer 1

The correct option is D

p - 4, q - 1, r - 3, s - 2

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

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

We see, both x and y intercepts are positive. a> 0, b > 0

By definition of sin$\theta$, cos$\theta$, tan$\theta$, cot$\theta$, sec$\theta$, and cosec$\theta$

All the trigonometric function gives positive values in first quadrant.

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

We see that x intercept in second quadrant is

negative. Therefore,a <0,b>0.

Functions which includes the values of

x - Intercept or adjacent will give negatives values.

All the other functions will give positive values.

By definition we know except sin$\theta$ & cosec$\theta$ , all the other functions use adjacent in their trigonometric ratios.

So, only sin$\theta$ & cosec$\theta$ give positive values in second quadrant.

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

We see both x& y - intercepts are negative

in III quadrant.

Therefor a < 0, b < 0

Both adjacent and opposite are negative.

So, trigonometric ratio which uses both adjacent and opposite gives positive value.

Only tan$\theta$ & cot$\theta$ use both adjacent and opposite.

In III quadrant, only tan$\theta$ & cot$\theta$ give positive values

Remaining trigonometric functions will give negative values.

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

X intercept is positive while y- intercepts negative.

Therefore a > 0, b < 0

Function which uses the values of y- intercept gives negative values, remaining functions give positive value.

Only cos$\theta$ & sec$\theta$ doesn't will give positive value in fourth quadrant.