1) Verify that the x-values are solutions of the equation.
csc x - 2 = 0
a) x=pi/6 b)x=5pi/6
I know that csc x = 2 but I do not know what to do now.
If csc x = 2, sin x = 1/2.
The angles with sin = 1/2 are 30 degrees (pi/6) and 150 degrees (5 pi/6).
To verify whether the given x-values are solutions of the equation csc x - 2 = 0, we need to substitute each value into the equation and check if it satisfies it. Here's how you can do it:
a) x = pi/6:
Substituting x = pi/6 into the equation, we have:
csc(pi/6) - 2 = 0
Now, you need to find the value of csc(pi/6), which is the reciprocal of sin(pi/6).
sin(pi/6) = 1/2
Reciprocal of 1/2 is 2. So, csc(pi/6) = 2.
Substituting back into the equation, we have:
2 - 2 = 0
Since 0 = 0, the equation holds true.
Therefore, x = pi/6 is a solution.
b) x = 5pi/6:
Substituting x = 5pi/6 into the equation, we have:
csc(5pi/6) - 2 = 0
Now, find the value of csc(5pi/6), which is the reciprocal of sin(5pi/6).
sin(5pi/6) = 1/2
Reciprocal of 1/2 is 2. So, csc(5pi/6) = 2.
Substituting back into the equation, we have:
2 - 2 = 0
Again, since 0 = 0, the equation holds true.
Therefore, x = 5pi/6 is also a solution.
In conclusion, both a) x = pi/6 and b) x = 5pi/6 are solutions to the equation csc x - 2 = 0, as substituting these values into the equation results in a true statement.