Question

Solve the equation in the interval [0, 360).
csc theta = 1+ cot theta

Answer 1

let me replace theta with x for ease of typing

csc(x) = 1 + cot(x)
1/sin(x) = 1 + cos(x)/sin(x) by definition
1 = sin(x) + cos(x) by multiplying both sides by sin(x)
1 = sqrt(2) * sin(x + 45) by auxillary angle method
1/sqrt(2) = sin(x + 45)

we know that sin(45) = 1/sqrt(2), and also sin(90 + 45) = 1/sqrt(2)
also sin(360 + 45) = 1/sqrt(2)

lets see...

sin(x + 45) = sin(45), sin(90 + 45), sin(360 + 45)
therefore x = 0, 90, 360

however, you said the interval does not include 360...therefore reject that value
also notice that we had 1/sin(x) which means that sin(x) cannot equal 0.

however at x = 0, sin(x) = 0, but not at x = 90.

Therefore the only solution is x = 90 degrees

Answer 2

Or in your case theta

Answer 3

Thank you so much for your help! :)

Answer 4

Anytime