7.04 Graphical solutions to trigonometric equations

Find solutions graphically for sine/cosine equations up to a sin(bx)+c (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 grid lines, 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

question 1

question 2

question 3

 

Find solutions graphically for sine/cosine equations up to a sin (bx) + c (radians)

Solutions to an equation exist at the points where the curve crosses the $x$x axis.  

When we say find the solution to $y=f(x)$y=f(x), we are asking find all the values that make $f(x)=0$f(x)=0. 

Some functions have one solution; like linear, exponential and log functions

some have $2$2 solutions like quadratics 

and some may have an infinite number of solutions like the periodic functions we are currently studying. 

It is important to note that periodic functions that have been vertically translated may not have any solutions as they do not cross the $x$x-axis at all.

Solutions to a cyclic curve using a graph can be done by reading off the $x$x-intercepts (roots/solutions) from the graph within the required domain that suits the context or given requirements.

Worked examples

Question 4

Using a graph, state the $x$x-intercepts for $y=2\sin x$y=2sinx over the domain $-2\pi−2π<x<2π

Think:

Firstly we need to graph the curve, we can do this by hand using the steps outlined in the chapter on graphing sine curves, or we could use technology.  

Secondly identify the solutions, (the $x$x-intercepts) within the required domain. Note that the domain required for this question was not including the $-2\pi$−2π and $2\pi$2π, just the solutions between.

Do:

Once the solutions are identified read off the values and list them as solutions to the curve.

For this graph and function we have $3$3 solutions and they are $x=-\pi,x=0,x=\pi$x=−π,x=0,x=π

 

Question 5

State the $x$x-intercepts that occur over the domain $0\le x\le3\pi$0≤x≤3π  for the following function:

$y=-\cos\left(\frac{x}{2}-\frac{\pi}{2}\right)$y=−cos(x2​−π2​)

Think: Sketch the graph and mark the solutions that occur in the required domain.

Do:

The solutions to $y=-\cos\left(\frac{x}{2}-\frac{\pi}{2}\right)$y=−cos(x2​−π2​) over $0\le x\le3\pi$0≤x≤3π are $x=0$x=0 and $x=2\pi$x=2π

 

Reflect: We can also find the solutions to other types of functions for example, by plotting $y=f(x)$y=f(x) and the line $y=g(x)$y=g(x), we will be able to identify graphically the solution(s) to where $f(x)=g(x)$f(x)=g(x).  

Practice questions

Question 6

Question 7

Question 8

 

Analyzing solutions to trigonometric functions

While trigonometric equations can often be solved algebraically, it may be more practical to find approximate solutions by numerical methods including the use of graphic calculators and other software. In some cases, an algebraic solution is not possible and an approximation is the only option.

Trigonometric equations typically have many solutions due to the periodic nature of trigonometric functions. Care is needed to be sure that all the solutions over a given domain are found.

Worked example

Question 9

Given that $\tan x=\sin3x+5$tanx=sin3x+5, find all the values of $x$x that satisfy this equation between $0^\circ$0° and $360^\circ$360°.

Think: When we use a graphic calculator to find the solutions, we must check the mode to ensure it is set to the angle measurement used in the question. For this question, the mode should be set to degrees, 

Do: In the image below we have drawn the graphs of two functions, $f\left(x\right)=\tan x$f(x)=tanx and $g\left(x\right)=\sin3x+5$g(x)=sin3x+5. The region $0^\circ\le x\le360^\circ$0°≤x≤360° has been shaded, since we are only concerned about the solutions in this interval.

$f\left(x\right)=\tan x$f(x)=tanx (green) and $g\left(x\right)=\sin3x+5$g(x)=sin3x+5 (blue).

 

We can see that there are two points of intersection where $f\left(x\right)=g\left(x\right)$f(x)=g(x), and the $x$x-coordinates of these two points correspond to the solutions to the equation $\tan x=\sin3x+5$tanx=sin3x+5.

Using the appropriate functions on a graphing calculator, we determine that the points of intersection are $\left(76.707^\circ,4.233\right)$(76.707°,4.233) and $\left(260.340^\circ,5.875\right)$(260.340°,5.875). So the solutions to the equation $\tan x=\sin3x+5$tanx=sin3x+5 are $x=77^\circ$x=77° and $x=260^\circ$x=260°, to the nearest degree.

 

Question 10

Given that $\cot\frac{x}{2}=3$cotx2​=3, find all values of $x$x between $-2\pi$−2π and $2\pi$2π.

Think: The equation has the trigonometric function isolated; therefore we can focus on determining the inverse of cotangent. A calculator will normally have the inverse tangent function but probably not the inverse cotangent function.

Do: A graphical method for this problem would be to plot on the same axes the functions $y=\cot\frac{x}{2}$y=cotx2​ and $y=3$y=3. The solutions we seek are at the intersection points of these graphs. Usually, a graphing calculator will be capable of finding the intersections automatically. The display could look something like the following.

Using technology, we would find that $x=-5.64$x=−5.64 and $x=0.64$x=0.64 are the two solutions within the specified interval, $-2\pi$−2π to $2\pi$2π.

 

Exploration

Some equations have no algebraic methods of solution. Suppose we have $e^{x-4}=\sin(2x)$ex−4=sin(2x). On a graphing calculator, we could plot the left- and right-hand sides of the equation as separate functions and determine the intersection points. The display looks something like the following.

We can see that there are four solutions between $x=0$x=0 and $x=2\pi$x=2π.

By calculator, we obtain these approximate intersection points:
$\left(0.009,0.018\right)$(0.009,0.018)
$\left(1.53,0.084\right)$(1.53,0.084)
$\left(3.45,0.576\right)$(3.45,0.576)
$\left(3.99,0.992\right)$(3.99,0.992)

The solutions to the equation $e^{x-4}=\sin(2x)$ex−4=sin(2x) are the $x$x-coordinates of these points of intersection.

 

Practice questions

Question 11

Question 12

Question 13

Question 14

Question 15