We obtained the value for cos θ. Recall that the cosine of an angle in standard position is the x-coordinate of the point of intersection P of its terminal side and the unit circle.
P(x,y)=(cosθ,sinθ)
To solve the equation cos θ = 12, we need to consider the points on the unit circle that have a x-coordinate of 12. Recall that cosine is positive in Quadrants I and IV.
We can now draw two congruent right triangles, each with a leg on the x-axis. Since the radius of the unit circle is 1, the hypotenuse of each triangle is also 1. Furthermore, since the x-coordinate of both points is 12, the length of the leg that is on the x-axis of each triangle is 12.
For both right triangles, one of the legs is 12 times the hypotenuse, so they are 30^(∘)-60^(∘)-90^(∘) triangles. In this type of triangle, the middle angle measures 60^(∘), or π3 radians. Knowing that a full turn measures 2π radians, we can calculate the desired angle measures. Since π3 is in Quadrant I, it is already simplified. Therefore, we will subtract π3 from 2π to get the second angle.
We found two solutions for the equation cos θ = 12. These are also the solutions for the given equation.
θ= π3 radians and θ= 5π3 radians
