 Hong Kong
Stage 4 - Stage 5

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

Lesson

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°

Careful!

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 $y=-4\sin x$y=4sinx and $y=-4$y=4 are drawn below.

1. State all solutions to the equation $-4\sin x=-4$4sinx=4 over the domain $\left[-360^\circ,360^\circ\right]$[360°,360°]. Give your answers in degrees separated by commas.

##### question 2

Consider the function $y=\cos3x$y=cos3x.

1. Draw the function $y=\cos3x$y=cos3x.

2. Draw the line $y=0.5$y=0.5 below.

3. Hence, state all solutions to the equation $\cos3x=0.5$cos3x=0.5 over the domain $\left[-60^\circ,60^\circ\right]$[60°,60°]. Give your answers in degrees separated by commas.

##### question 3

Consider the function $y=2\sin2x$y=2sin2x.

1. Draw the function $y=2\sin2x$y=2sin2x.

2. State the other function you would draw in order to solve the equation $2\sin2x=1$2sin2x=1 graphically.
3. Draw the line $y=1$y=1 below.
4. Hence, state all solutions to the equation $2\sin2x=1$2sin2x=1 over the domain $\left[-180^\circ,180^\circ\right]$[180°,180°]. Give your answers in degrees separated by commas.