Find solutions from a graph for sine and cosine equations (degrees)

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 $\sin x=\frac{1}{2}$sinx=12​. Graphically speaking, this is the same as finding the $x$x-values that correspond to the points of intersection of the curves $y=\sin x$y=sinx and $y=\frac{1}{2}$y=12​.

$y=\sin x$y=sinx (blue) and $y=\frac{1}{2}$y=12​ (orange).

 

We can see in the region given by $\left(-360^\circ,360^\circ\right)$(−360°,360°) that there are four points where the two functions meet.

Red 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 $30^\circ$30°, which means that the solution to the equation $\sin x=\frac{1}{2}$sinx=12​ is given by:

$x=-330^\circ,-210^\circ,30^\circ,150^\circ$x=−330°,−210°,30°,150°

Practice questions

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