Question

Determine which quadrant the angle Ф lies and the reference angle.

Then, find the indicated ratio for Ф' (must be an exact value).
Finally, determine the value for the original expression using the quadrants.

csc
`\cfrac{4\pi }{3}`

Answer 1

`\\ \rm\Rrightarrow csc(4\pi)/(3)`

`\\ \rm\Rrightarrow csc\left(\pi +(\pi)/(3)\right)`

• So it lies in Q3 and it's reference angle is π/3
• Reference angle like usual lies in Q1 .

Now for value

`\\ \rm\Rrightarrow csc\left(\pi+(\pi)/(3)\right)`

• In Q1 All are positive
• In Q2 sine and cosec are positive .
• In. Q3 tan and cot are positive .
• In Q4 cos and sec are positive.

As it lies in Q3 it's negative

`\\ \rm\Rrightarrow csc\left(-(\pi)/(3)\right)`

`\\ \rm\Rrightarrow -csc(\pi)/(3)`

`\\ \rm\Rrightarrow -(2)/(√(3))`

Answer 2

`\csc \left((4\pi)/(3)\right)=-(2)/(√(3))`

Explanation:

`\phi=(4 \pi)/(3)`

Therefore, this angle lies in quadrant III since it is between π and 3π/2

It's reference angle is:

`(4 \pi)/(3)-\pi=(\pi)/(3)`

and so lies in quadrant I

`\csc (\phi)=(1)/(\sin(\phi))`

For sine, quadrant I and II are positive and quadrant III and IV are negative.

Therefore, as
`\sin (\pi)/(3)=(√(3))/(2)` then
`\sin (4\pi)/(3)=-(√(3))/(2)`

Finally,

`\csc \left((4\pi)/(3)\right)=(1)/(\sin\left((4\pi)/(3)\right))`

`\implies \csc \left((4\pi)/(3)\right)=(1)/(-(√(3))/(2))`

`\implies \csc \left((4\pi)/(3)\right)=-(2)/(√(3))`