  # Cosine Equation Solver

 

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 $2\pi$ ).

## Graphical interpretations of the solutions of the equation $\cos(x)= a$ over the interval $[ 0, 2\pi)$

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 $\cos(x) = a$. ## Analytical Solutions of the equation $\cos(x)= a$ over the interval $[ 0, 2\pi)$

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 $x_2 = \pi + x_r$

## Use the calculator to solve the equation $\cos(x)= a$ for $x$ over the interval $[ 0, 2\pi)$

Enter $a$ as a real number and press "enter". You may also set the "Number of Decimal Places" desired as a positive integer.

 $a$ = 0.5 Number of Decimal Places = 3