Equation (I) resulted in a false statement. This means that Equation (I) has no solutions, and we can therefore discard it and fully focus on Equation (II).
The cosine of an angle in the standard position is the x-coordinate of the point of intersection P at which the terminal side of the angle intersects the unit circle.
P(x,y)=(cosθ,sinθ)
We want to find the values of θ that are the solutions to our equation, cos θ =- 1. To do so, we need to consider the points on the unit circle that have the x-coordinate of - 1, a negative value. Recall that cosine is negative in Quadrants II and III.
Notice that there is only one point on a unit circle that has - 1 as the x-coordinate.
Since we know that a half turn measure is 180 ^(∘), we have found one solution of the equation cos θ=- 1.
θ= 180 ^(∘)
Finally, keep in mind that if we add or subtract a multiple of 360^(∘), the terminal side of the angle will be in the same position. This means that the cosine of the resulting angles will also be - 1. Therefore, we can now write all angles which cosine is -1.
θ= 180 ^(∘) + n * 360^(∘)
where n is any integer
All angles fulfilling the formula are solutions to our original equation.
