One solution to sinθ=22 is θ=4π. To find a second solution to this equation, we need to use the unit circle. Remember, sine is positive in the first and second quadrants of the coordinate plane.
The sine of an angle in standard position is the y-coordinate of the point of intersection of the angle's terminal side and the unit circle. The solution θ=4π is in Quadrant I, so the second solution must be in Quadrant II. Because these angles are symmetric across the y-axis and half a turn measures π radians, to find this angle we subtract 4π from π.
Another solution to the equation sinθ=22 is θ=43π.
sinθ=-22
To find a solution to this equation, we will use an inverse trigonometric function and a calculator.
One solution to sinθ=-22 is θ=-4π. To find a second solution to this equation, we need to use the unit circle. Remember, sine is negative in the third and fourth quadrants of the coordinate plane.
The solution θ=-4π is in Quadrant IV, so the second solution must be in Quadrant III, and these angles are symmetric across the y-axis. Knowing that half a turn measures π radians and a full turn measures 2π radians, we can calculate the desired angle measures. We will add 4π to π, and subtract 4π from 2π.
The solutions to the equation sinθ=-22 are θ=45π and θ=47π.
x-intercepts
We found six solutions to the given equation, θ=0,θ=π,θ=4π,θ=43π,θ=45π, and θ=47π. 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.
Note that the first two solutions differ by π, and π is half of 2π. Also, the other four solutions differ from each other by 2π, and 2π is a quarter of 2π. With this information in mind, we can write our answer in a simpler form.
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