To solve this equation, we first rewrite it using the Pythagorean identity:
The Pythagorean identity is cos²θ + sin²θ = 1.
From there, we can rewrite the equation from cos²θ - sin²θ = 1 - sinθ, to cos²θ = 1 - sinθ + sin²θ.
Then, we simplify it to cos²θ = (1 - sinθ)².
This equation is then possible when cosθ = 1 - sinθ or cosθ = -(1 - sinθ).
Splitting the equation in this way, we have two resulting possible solutions:
Equation 1) cosθ = 1 - sinθ.
Equation 2) cosθ = -1 + sinθ, which can be rewritten to: cosθ = sinθ - 1.
For Equation 1) Using the geometric interpretation: y=cosθ, x=sinθ, and r=1, (From the definition of the Trigonometric functions with a unit circle)
We have y = 1-x. This defines a straight line intersecting the unit circle.
For Equation 2) Similarly, we also have a straight line defined by y = x - 1 intersecting with the unit circle at some points.
Solving for the intersections (values of theta) will yield four intersections in total, two for each equation.
Through the graphical interpretation, the values can then be calculated. Points of intersection will give the appropriate values for theta in their respective intervals in the unit circle. This involves understanding the graph of the unit circle and the straight line as well as converting point coordinates back to angles using inverse trigonometric functions.
After calculating the angles, we need to filter out those that are outside of the given interval of 0 to 2π. This gives us the specific solutions to the equation cos²θ - sin²θ = 1 - sinθ.