Find all solutions to the equation 1 = 2cos θ.
well, you know the solutions will all be in QI and QIV
and you know cosθ = 1/2
To find all solutions to the equation 1 = 2cos θ, we can start by rearranging it to isolate the cosine term:
2cos θ = 1
Divide both sides of the equation by 2:
cos θ = 1/2
Now, to find the solutions for θ, we need to think about the trigonometric values of the angle whose cosine is 1/2.
We know that the cosine function gives the ratio of the adjacent side to the hypotenuse in a right triangle. Therefore, we need to find the angles where the ratio of the adjacent side to the hypotenuse is 1/2.
One way to find these angles is by using the unit circle. On the unit circle, the x-coordinate represents the cosine of an angle. We need to find the angles where the x-coordinate is 1/2.
The unit circle shows that there are two angles, θ1 and θ2, that satisfy cos θ = 1/2:
θ1 = π/3 (or 60 degrees)
θ2 = 5π/3 (or 300 degrees)
Notice that we can find all solutions to the equation by adding or subtracting multiples of 2π (or 360 degrees) to these two angles, since the cosine function is periodic with a period of 2π.
Therefore, the solutions to the equation 1 = 2cos θ are:
θ1 = π/3 + 2πn, where n is an integer
θ2 = 5π/3 + 2πn, where n is an integer
These equations give an infinite number of solutions, as we can select any integer value for n and find a corresponding θ.