An online calculator to solve simple cosine equations of the form \( \cos x = a \) is presented.
Solution on the interval \( [ 0, 2\pi) \) or \( [ 0, 360^{\circ}) \) are presented. (Note that the interval is open at \( 2\pi \) ).
Below is shown the graph of \( y = \cos(x) \) and \( y = a \) (horizontal lines in red) and it can be seen that there are no points of intersection of the two graphs for \( a \gt 1 \) or \( a \lt - 1 \) and therefore the equation \( \cos(x) = a \) has no solution.
For \( a = 1 \), there is one point of tangency \( A \) hence one solution given by \( x = 0 \)
For \( a = - 1 \), there is one point of tangency \( H \) hence one solution given by \( x = \pi \)
There are two points of intersections for all the remaining values of \( a \) and hence there are two solutions.
The calculator presented below allows you to practice different cases, as described above, and hence fully understand how to solve equations of the form \( \cos(x) = a \).
1) The range of \( y = \cos(x) \) is given by the interval \( [ 1, -1] \) (see graph above). Therefore \( \cos(x) \) cannot take values outside the interval \( [ 1, -1] \) and therefore the equation \(\cos(x) = a \) has no real solution for \( a \gt 1 \) or \( a \lt -1 \).
2) The solution to the equation \(\cos(x) = 1 \) is given by: \( x_1 = 0 \)
3) The solution to the equation \(\cos(x) = -1 \) is given by: \( x_1 = \pi \)
4) The solutions to the equation \(\cos(x) = 0 \) are given by: \( x_1 = \dfrac{\pi}{2} \) and \( x_2 = \dfrac{3\pi}{2} \)
5) For all other values of \( a \), we solve the equation \(\cos(x) = a \) as follows:
Let the reference angle \( x_r = \arccos(|a|) \)
If \( 0 \lt a \lt 1\) , there are two solutions given by: \( x_1 = x_r\) and \( x_2 = 2\pi - x_r \)
If \( -1 \lt a \lt 0\) , there are two solutions given by: \( x_1 = \pi - x_r + \pi\) and \( x_2 = \pi + x_r \)
