Can someone please help me solve this equation?
Solve the equation in the interval [0°, 360°).
csc θ = 1 + cot θ
1/sinθ = 1 + cosθ/sinθ
1 = sinθ + cosθ
1 - sinθ = cosθ
1-2sinθ + sin^2 θ = cos^2 θ
1 - 2sinθ + sin^2 θ = 1 - sin^2 θ
2sin^2 θ - 2sinθ = 0
2sinθ(sinθ - 1) = 0
sinθ = 0, θ = 0,180,360°
or sinθ = 1 , θ = 90°
but,...
since we squared, all our answers must be verified.
since csc x is undefined for x = 0,180, 360 none of those will work
so x = 90°
To solve the given equation, csc θ = 1 + cot θ, in the interval [0°, 360°), we can use trigonometric identities and algebraic manipulation.
We'll start by rewriting the equation using the reciprocal identities:
csc θ = 1 + cot θ
1/sin θ = 1 + cos θ/sin θ
Next, we'll multiply both sides of the equation by sin θ to eliminate the denominators:
1 = sin θ + cos θ
Now, we can square both sides of the equation to eliminate the square root:
1^2 = (sin θ + cos θ)^2
1 = sin^2 θ + 2sin θcos θ + cos^2 θ
Using the Pythagorean identity sin^2 θ + cos^2 θ = 1, we can simplify the equation further:
1 = 1 + 2sin θcos θ
0 = 2sin θcos θ
Now, we have two possibilities:
1. sin θ = 0
2. cos θ = 0
For sin θ = 0, we know that sin θ equals 0 at θ = 0° and θ = 180°.
For cos θ = 0, we know that cos θ equals 0 at θ = 90° and θ = 270°.
Therefore, the solutions to the equation csc θ = 1 + cot θ in the interval [0°, 360°) are θ = 0°, 90°, 180°, and 270°.