math problem?

Question

What can you do to solve this problem:

find the exact value of csc 5pie/3

Answer 1

The cosecant is 1 over the sin, by definition:

csc(x) = 1/sin(x)

So

csc(5π/3)

becomes

1/sin(5π/3)

5π/3 is just π/3 less than 6π/3 = 2π, so that makes it a 60° in the 4th quadrant, where sin is negative. Since sin(π/3) = √3/2, sin(5π/3) = -√3/2.

= 1/(-√3/2)
= -2/√3

And then rationalize by multiplying numerator and denominator by √3/√3 to get:

= -2√3/3

Answer 2

21

Answer 3

You can put it in your calculator. Csc is 1/sin, so put in 1/sin(5pi/3).

Answer 4

csc(5pi/3)

= 1 / sin(5pi/3)
because of formula : csc(A) = 1 / sin(A).

= 1 / sin(pi + 2pi/3)
by dividing 3 into 5pi.

=1 / [sin(pi)cos(2pi/3) + sin(2pi/3)cos(pi)]
because of formula : sin(A + B) = sin(A)cos(B) + sin(B)cos(A).

= 1 / [0 - sin(2pi/3)]
because sin(pi) = 0 and cos(pi) = -1.

= -1 / sin(2pi/3)

= -1 / [sqrt(3) / 2]
from tables of sin values.

= -2 / sqrt(3)

= [-2 / sqrt(3)] * [sqrt(3) / sqrt(3)]

= -2 * sqrt(3) / 3

Answer 5

879