SOLUTION: Find all exact solutions on the interval 0 ≤ x < 2𝜋. (Enter your answers as a comma-separated list.) csc2(x) − 7 = −5

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Question 1203693: Find all exact solutions on the interval
0 ≤ x < 2𝜋. (Enter your answers as a comma-separated list.)
csc2(x) − 7 = −5
Found 2 solutions by MathLover1, math_tutor2020:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

exact solutions
csc%5E2%28x%29+-7+=+-5
on the interval
0+%3C=+x+%3C+2pi

use identity csc%5E2%28x%29+=1%2Fsin%5E2%28x%29

1%2Fsin%5E2%28x%29+-7+=+-5
1%2Fsin%5E2%28x%29++=+-5%2B7
1%2Fsin%5E2%28x%29++=2
sin%5E2%28x%29++=1%2F2
sin%28x%29++=sqrt%281%2F2%29

solutions:
x=pi%2F4%2B2pi%2An
x=3pi%2F4%2B2pi%2An

on the interval 0+%3C=x+%3C+2pi
x=pi%2F4
x=3pi%2F4

answer: pi%2F4 , 3pi%2F4




Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Tutor @MathLover1 has the right idea.
However, keep in mind that sin%5E2%28x%29+=+1%2F2 leads to sin%28x%29+=+%22%22+%2B-+sqrt%281%2F2%29.
Don't forget about the plus minus.

It's similar to how x%5E2+=+9 has solutions x = -3 and x = 3.

sin%28x%29+=+%22%22+%2B-+sqrt%281%2F2%29 breaks down into
Those 2 new equations have 2 solutions each on the interval 0+%3C=+x+%3C+2pi

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Ultimately there are four solutions to csc%5E2%28x%29-7=-5 on the interval 0+%3C=+x+%3C+2pi
Those four solutions are:
pi/4
3pi/4
5pi/4
7pi/4

The numerators have coefficients 1,3,5,7 to give a nice pattern.
Those angles correspond to 45°, 135°, 225°, 315°
Adjacent angles are separated by 90° (aka pi radians)
I recommend using the unit circle.