Find the exact value of csc 4pi/3
What I have determined thus far:
4pi/3 = 60 degrees
sin = square root of 3/2
csc=1/sin=1/sqrt(3)/2)=
= 2/sqrt3
And the exact value for csc4pi/3 is determined how?
-2sqrt3/3?
actually, 4pi/3 = pi + pi/3,
csc 4pi/3 = -csc pi/3 = -2/√3
or, if you insist on rational denominators, -2√3/3
To find the exact value of csc(4π/3), we can start by finding the value of sin(4π/3) first.
We know that 4π/3 is equivalent to 240 degrees. In this quadrant, the sine function is negative.
By evaluating sin(240 degrees), we can see that sin(4π/3) is equal to -√3/2.
Now, to find the cosecant (csc) of an angle, we can use the reciprocal property of trigonometric functions.
Therefore, csc(4π/3) = 1/sin(4π/3).
Plugging in the value we just found, we have csc(4π/3) = 1/(-√3/2).
To simplify this expression, we multiply the numerator and denominator by 2 to get csc(4π/3) = -2/√3.
However, it is a convention to rationalize the denominator, so we will multiply both the numerator and denominator by √3 to get csc(4π/3) = -2√3/3.
Therefore, the exact value of csc(4π/3) is -2√3/3.