Trigonometric graphs in radians
When sketching trigonometric functions using radians, we can simply change the values on the horizontal axis from degrees to radians. The domain also changes accordingly.
Now that we are working in radians it is easier to graph other functions together with trigonometric functions in order to solve equations graphically. This is because we are no longer using degrees but radians on the $x$x-axis, meaning that other functions of $x$x are possible.
Worked example
Example 1
Determine the number of solutions that satisfy the equation $2\sin x=x$2sinx=x.
Think: What does the graph of $y=2\sin x$y=2sinx and $y=x$y=x look like? The solutions to the equation will be the same as the points of intersection of the two graphs.
Do: Draw a number plane and mark a scale with $\frac{-\pi}{2}$−π2, $-\pi$−π, $\frac{\pi}{2}$π2, $\pi$πetc. and also approximate the positions of $-1$−1,$-2$−2,$0$0,$1$1,$2$2, and $3$3 etc. (you can do this using your calculator: $pi\approx3.14$pi≈3.14).
Sketch the graph of $y=2\sin x$y=2sinx and $y=x$y=xand look for the number of times the graphs meet.
It is clear that the graphs meet at three points, therefore there are $3$3 solutions to the equation $2\sin x=x$2sinx=x.
Practice questions
Question 1
Consider the graph of $y=\sin x$y=sinx given below.
Using the graph, what is the sign of $\sin\frac{13\pi}{12}$sin13π12?
Positive
ANegative
BWhich quadrant does an angle with measure $\frac{13\pi}{12}$13π12 lie in?
Quadrant $I$I
AQuadrant $II$II
BQuadrant $IV$IV
CQuadrant $III$III
D
Question 2
Consider the identity $\sec x=\frac{1}{\cos x}$secx=1cosx and the table of values below.
| $x$x | $0$0 | $\frac{\pi}{4}$π4 | $\frac{\pi}{2}$π2 | $\frac{3\pi}{4}$3π4 | $\pi$π | $\frac{5\pi}{4}$5π4 | $\frac{3\pi}{2}$3π2 | $\frac{7\pi}{4}$7π4 | $2\pi$2π |
|---|---|---|---|---|---|---|---|---|---|
| $\cos x$cosx | $1$1 | $\frac{1}{\sqrt{2}}$1√2 | $0$0 | $-\frac{1}{\sqrt{2}}$−1√2 | $-1$−1 | $-\frac{1}{\sqrt{2}}$−1√2 | $0$0 | $\frac{1}{\sqrt{2}}$1√2 | $1$1 |
For which values of $x$x in the interval $\left[0,2\pi\right]$[0,2π] is $\sec x$secx not defined?
Write all $x$x-values on the same line separated by commas.
Complete the table of values:
$x$x $0$0 $\frac{\pi}{4}$π4 $\frac{\pi}{2}$π2 $\frac{3\pi}{4}$3π4 $\pi$π $\frac{5\pi}{4}$5π4 $\frac{3\pi}{2}$3π2 $\frac{7\pi}{4}$7π4 $2\pi$2π $\sec x$secx $\editable{}$ $\editable{}$ undefined $\editable{}$ $\editable{}$ $\editable{}$ undefined $\editable{}$ $\editable{}$ What is the minimum positive value of $\sec x$secx?
What is the maximum negative value of $\sec x$secx?
Plot the graph of $y=\sec x$y=secx on the same set of axes as $y=\cos x$y=cosx.
Loading Graph...
Question 3
Consider the graph of $y=5\sin x$y=5sinx given below.
Plot the graph of $y=x$y=x.
Loading Graph...From part (a), determine how many solutions there are to the equation $5\sin x=x$5sinx=x.