How do you find two solutions (in degree and radians) for cscx = (2sqrt3)/3?

Question

Quinn Anderson · Accepted Answer

Solve #csc x = (2sqrt3)/3#

Ans: #pi/3 and (2pi)/3# #csc x = 1/(sin x) = (2sqrt3)/3.# Find sin x. #sin x = 3/(2sqrt3) = sqrt3/2.# Trig Table of Special Arcs gives --> #sin x = sqrt3/2# ---> arc #x = pi/3 (or 60^@)#, and #x = (2pi)/3 (or 120^@)#

Kai Anderson · Answer

undefined To find the solutions for the equation csc(x) = (2√3)/3, you can follow these steps:

1. Identify the reference angle θ in the first quadrant using the given value: csc(θ) = (2√3)/3. The reference angle is the angle whose sine is equal to the reciprocal of the given value. In this case, θ = 30 degrees or π/6 radians.

2. Recognize that csc(x) = 1/sin(x), so if csc(x) = (2√3)/3, then sin(x) = 3/(2√3).

3. Since sin(x) = 3/(2√3), we can find the angle x by taking the inverse sine (arcsin) of 3/(2√3).

4. Calculate the values of x using the inverse sine function. Remember that sine is positive in the first and second quadrants.

5. Once you find the value of x in radians, convert it to degrees if necessary.

So, the solutions in degrees and radians are:

1. $ x = 30^\circ $ (or $ x = \frac{\pi}{6} $ radians)
2. $ x = 180^\circ - 30^\circ $ (or $ x = \pi - \frac{\pi}{6} $ radians)