Graphical solution of trigonometric equations involving sine and cosine
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
some have $3$3 solutions like cubics
and some may have an infinite number of solutions like the cyclic functions we are currently studying.
Some functions, however, have no solutions,
Like linear functions that are horizontal lines
Quadratic functions with vertical translations
These absolute value functions
And also some cyclic functions. Cyclic 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.
Examples
Example 1
Using a graph, state the solutions to the following sine curve ($2\sin x$2sinx) over the domain $-2\pi
Think:
Firstly I need to graph the curve, I can do this by hand using the steps outlined in the chapter on graphing sine curves, or I 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 $x=-\pi,0,\pi$x=−π,0,π
Example 2
By first sketching the graph of $y=-\cos\left(\frac{x}{2}-\frac{\pi}{2}\right)$y=−cos(x2−π2), state the solutions that occur over the domain $0\le x\le3\pi$0≤x≤3π
After sketching the graph and marking on the solutions that occur in the required domain, we just now need to list the solutions.
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
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). See the following examples.
Worked Examples
Question 1
Consider the function $y=3\sin x$y=3sinx.
Graph this function.
Loading Graph...Add the line $y=3$y=3 to your graph.
Loading Graph...Hence, state all solutions to the equation $3\sin x=3$3sinx=3 over the domain $\left[-2\pi,2\pi\right]$[−2π,2π]. Give your answers as exact values separated by commas.
Question 2
Consider the function $y=\cos\left(\frac{x}{4}\right)$y=cos(x4).
Graph this function.
Loading Graph...Add the line $y=-0.5$y=−0.5 to your graph.
Loading Graph...Hence, state all solutions to the equation $\cos\left(\frac{x}{4}\right)=-0.5$cos(x4)=−0.5 over the domain $\left[-4\pi,4\pi\right]$[−4π,4π]. Give your answers as exact values separated by commas.
Question 3
Consider the function $y=\tan\left(x-\frac{\pi}{4}\right)$y=tan(x−π4).
Graph this function.
Loading Graph...Add the line $y=1$y=1 to your graph.
Loading Graph...Hence, state all solutions to the equation $\tan\left(x-\frac{\pi}{4}\right)=1$tan(x−π4)=1 over the domain $\left[-2\pi,2\pi\right)$[−2π,2π). Give your answers as exact values separated by commas.