Find solutions from a graph for sine and cosine equations (radians)
The points of intersection of two curves, $y=f\left(x\right)$y=f(x) and $y=g\left(x\right)$y=g(x) can be obtained by first setting the expressions $f\left(x\right)$f(x) and $g\left(x\right)$g(x) to be equal and then solving for $x$x. More specifically, we want to find the values of $x$x that satisfy the equation:
$f(x)=g(x)$f(x)=g(x)
Then, naturally, we would substitute our $x$x-values into either $y=f\left(x\right)$y=f(x) or $y=g\left(x\right)$y=g(x) to find the corresponding $y$y-values of the points of intersection. In general, solving an equation can be thought of as finding the $x$x-values of the points of intersection of two curves.
Exploration
Say we wanted to find the values of $x$x that solve the equation $\sin\left(x-\frac{\pi}{3}\right)=1$sin(x−π3)=1. Graphically speaking, this is the same as finding the $x$x-values that correspond to the points of intersection of the curves $y=\sin\left(x-\frac{\pi}{3}\right)$y=sin(x−π3) and $y=1$y=1.
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| $y=\sin\left(x-\frac{\pi}{3}\right)$y=sin(x−π3) (green) and $y=1$y=1 (blue). |
We can see in the region given by $\left(-2\pi,2\pi\right)$(−2π,2π) that there are two points where the two functions meet.
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| Points indicating where the two functions meet. |
Since we are fortunate enough to have gridlines, the $x$x-values for these points of intersection can be easily deduced. Each grid line is separated by $\frac{\pi}{6}$π6, which means that the solution to the equation $\sin\left(x-\frac{\pi}{3}\right)=1$sin(x−π3)=1 in the region $\left(-2\pi,2\pi\right)$(−2π,2π) is given by:
$x=-\frac{7\pi}{6},\frac{5\pi}{6}$x=−7π6,5π6
We can only solve equations graphically if the curves are drawn accurately and to scale. You won't be expected to solve equations graphically if it requires drawing the curves by hand.
Practice questions
question 1
The functions $y=\sin\left(x-\frac{\pi}{6}\right)+4$y=sin(x−π6)+4 and $y=\frac{9}{2}$y=92 are drawn below.
State all solutions to the equation $\sin\left(x-\frac{\pi}{6}\right)+4=\frac{9}{2}$sin(x−π6)+4=92 over the domain $\left[-2\pi,2\pi\right)$[−2π,2π). Give your answers as exact values separated by commas.
question 2
Consider the functions $y=-\cos\left(x-\frac{\pi}{4}\right)-2$y=−cos(x−π4)−2 and $y=-3$y=−3.
Draw the functions $y=-\cos\left(x-\frac{\pi}{4}\right)-2$y=−cos(x−π4)−2 and $y=-3$y=−3.
Loading Graph...Hence, state all solutions to the equation $-\cos\left(x-\frac{\pi}{4}\right)-2=-3$−cos(x−π4)−2=−3 over the domain $\left(-2\pi,2\pi\right)$(−2π,2π). Give your answers as exact values separated by commas.
question 3
Consider the functions $y=2\sin\left(x-\frac{\pi}{6}\right)+5$y=2sin(x−π6)+5 and $y=6$y=6.
Draw the functions $y=2\sin\left(x-\frac{\pi}{6}\right)+5$y=2sin(x−π6)+5 and $y=6$y=6.
Loading Graph...Hence, state all solutions to the equation $2\sin\left(x-\frac{\pi}{6}\right)+5=6$2sin(x−π6)+5=6 over the domain $\left[-2\pi,2\pi\right)$[−2π,2π). Give your answers in radians, separated by commas.