The of the graph of a occur when
y=0. Let's substitute
0 for
y into the given equation.
y=2cosθ+1⇓0=2cosθ+1
We will start by solving the equation for
cosθ.
0=2cosθ+1
-1=2cosθ
2-1=cosθ
-21=cosθ
cosθ=-21
To find a solution for this equation, we will use an and a calculator.
cosθ=-21
θ=cos-1(-21)
θ=32π
One solution to
cosθ=-21 is
θ=32π. To find the second solution for this equation, we need to use the . Remember, is negative in the second and third of the coordinate plane.
The cosine of an angle in is the x-coordinate of the point of intersection of the unit circle and the of the angle. The solution θ=32π is in Quadrant II, so the second solution must be in Quadrant III. Because these angles are symmetric across the x-axis and a full turn measures 2π , to find this angle we subtract 32π from 2π.
We found two solutions to the given equation,
θ=32π and
θ=34π. Finally, keep in mind that if we add or subtract a multiple of
2π radians, the terminal side of the angle will be in the same position. Therefore, the resulting angles will also be solutions to the equation.
32π+2πn radians,34π+2πn radians,where n is any integer
These are the
x-intercepts of the graph of
y=2cosθ+1.