Question 1189670: Solve the following equations for x between 0° and 360°
a) cosec x = -2
b) sec x = 5 sin 20°
c) cot^2 x = 3 Found 3 solutions by Alan3354, MathLover1, Theo:Answer by Alan3354(69443) (Show Source):
You can put this solution on YOUR website! Solve the following equations for x between 0° and 360°
a) cosec x = -2
sin(x) = -1/2
x = 210, 330 degs
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b) sec x = 5 sin 20°
1/cos(x) = 5sin(20)
cos(x) = 1/(5sin(20)) = ???? use a calculator
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c) cot^2(x) = 3
tan^2(x) = 1/3
cosec(x) = 1/sin(x)
equation becomes:
1/sin(x) = -2
solve for sin(x) to get sin(x) = -1/2
sine is negative in third and fourth quadrant.
arcsin(1/2) = 30 degrees.
that's in the first quadrant.
equivalent angle in the third quadrant is 180 + 30 = 210 degrees.
equivalent angle in the fourth quadrant is 360 - 30 = 330 degrees.
x can either be 210 degrees or 330 degrees.
that's your solution.
here's what it looks like in a graph.
b) sec x = 5 sin 20°
sec(x) = 1/cos(x)
equation becomes:
1/cos(x) = 5sin(20)
solve for cos(x) to get:
cos(x) = 1/(5sin(20))
use your calculator to solve for 5sin(20).
you will get 5sin(20) = 1.710100717.
cos(x) = 1/that = .58476088.
solve for x to get:
x = 54.21390092 degrees.
that's in the first quadrant.
cosine is positive in the first and fourth quadrant.
equivalent angle in the fourth quadrant is 360 - 54.21390092 = 305.7860991 degrees.
x can be either 54.21390092 and 305.7860991 degrees.
that's your solution.
here's what it looks like on a graph.
c) cot^2 x = 3
cot^2(x) = 1/tan^2(x)
equation becomes:
1/tan^2(x) = 3.
solve for tan^2(x) to get:
tan^2(x) = 1/3.
solve for tan(x) to get:
tan(x) = sqrt(1/3)
solve for x to get:
x = arctan(sqrt(1/3)) = 30 degrees.
cotangent is positive in the first and third quadrant.
cotangent is negative in the second and fourth quadrant.
cotangent^2 is positive in all four quadrants.
equivalent angle in the second quadrant is 180 - 30 = 150 degrees.
equivalent angle in the third quadrant is 180 + 30 = 210 degrees.
equivalent angle in the fourth quadrant is 360 - 30 = 330 degrees.
x can be either 30, 150, 210, of 340 degrees.
that's your solution.
here's what it looks like on a graph.
