CAST diagram Maths

Question

Having never learned this and now in the thick of it for a degree, I am struggling with the below. This is an example of an exercise we have been given and with a written exam coming up end of this year i do not want to go in without a good understanding. I am assuming this will be a CAST diagram.

Solve the following,

𝑠𝑖𝑛𝜗−𝑐𝑜𝑡^2𝜗=1, express your solution in degrees in the range –180°≤𝜗≤180°, and in radians in the range 0°≤𝜗≤2πrad.

Thanks

Answer 1

sinx-cos²x/sin²x=1 ,,,,,definition of cotx

sin³x-cos²x=sin²x ,,,,,,,,,,,,,,multiply by sin²x both sides of equation

sin³x=cos²x+sin²x

sin³x=1 ,,,,,,basic trig ,,,,,,,,,cos²x+sin²x=1

given x is a real (not complex) number, sinx is real

only solution is

sinx=1

x=π/2+2πn ,,, n any interger or zero.

notice that when sinx or cosx equal +/-1, there is just one x angle value (+2πn)

that solves the equation.

as for CAST,,,, at angle x=π/2+2πn, sinx=1 and that is it maximum value reached once in a cycle

as for within range, n must be zero so x=π/2 or 90 degrees.