Question

cosec(7π/6)sec(5π/3)​

Answer 1

Answer:

since, cosec(7π/6) = -2

and sec(5π/3) = 2

= -2 × 2

= -4

Answer 2

Concept:

We need to know what is the value of cosec and sec in different quadrants.

Cosec∅ in:

1st quadrant= +ve

2nd quadrant= +ve

3rd quadrant= -ve

4th quadrant= -ve

Sec∅ in:

1st quadrant= +ve

2nd quadrant= -ve

3rd quadrant= -ve

4th quadrant= +ve

Also,

$cosec(\pi +x )=-cosec(x)$

$sec(2\pi -x})=sec(x )$

Given:

The trigonometric equation

$cosec(\frac{7\pi }{6} ).sec(\frac{5\pi }{3} )$

To find:

The value of $cosec(\frac{7\pi }{6} ).sec(\frac{5\pi }{3} )$

Solution:

Let's, find the value of

$cosec(\frac{7\pi }{6} )=cosec(\pi +\frac{\pi }{6} )$           [Lies in the 3rd quadrant]

⇒$cosec(\frac{7\pi }{6} )=-cosec(\frac{\pi }{6})$

⇒$cosec(\frac{7\pi }{6} )=-2$          --------------(1)

Next, we find the value of

$sec(\frac{5\pi }{3})=sec(2\pi -\frac{\pi }{3} )$               [Lies in the 4th quadrant]

⇒$sec(\frac{5\pi }{3})=sec(\frac{\pi }{3} )$

⇒$sec(\frac{5\pi }{3})=2$             -----------------(2)

From equations (1) and (2)

$cosec(\frac{7\pi }{6} ).sec(\frac{5\pi }{3} )=(-2)(2)$

⇒$cosec(\frac{7\pi }{6} ).sec(\frac{5\pi }{3} )=-4$

Therefore the value of

$cosec(\frac{7\pi }{6} ).sec(\frac{5\pi }{3} )$ is -4