The definitions of the trigonometric functions $\sin$sin, $\cos$cos and $\tan$tan that we've seen use the ratios of side lengths of a right-angled triangle. More specifically, we call this the right-angled triangle definition of the trigonometric functions, but there are other methods to define these functions more broadly.
For a right-angled triangle, where $\theta$θ is the measure for one of the angles (excluding the right angle), we have that:
$\sin\theta$sinθ | $=$= | $\frac{\text{opposite }}{\text{hypotenuse }}$opposite hypotenuse |
$\cos\theta$cosθ | $=$= | $\frac{\text{adjacent }}{\text{hypotenuse }}$adjacent hypotenuse |
$\tan\theta$tanθ | $=$= | $\frac{\text{opposite }}{\text{adjacent }}$opposite adjacent |
Now consider a right-angled triangle, with hypotenuse that has a length of one unit with a vertex centred at the origin. We can construct a unit circle around the triangle as shown below.
A right-angled triangle inscribed in the unit circle. |
The point indicated on the circle has coordinates $\left(\cos\theta,\sin\theta\right)$(cosθ,sinθ) using the right-angled triangle definition of $\cos$cos and $\sin$sin. Unfortunately, this definition is limited to angles with measures in the range of $0\le\theta\le\frac{\pi}{2}$0≤θ≤π2. However, more broadly, we can use the unit circle to define $\cos$cos and $\sin$sin for angles with any measure. We call this the unit circle definition. In this definition, the value of these functions will be the $x$x- and $y$y-values of a point on the unit circle after having rotated by an angle of measure $\theta$θ in the anticlockwise direction as shown below. If $\theta$θ is negative then the point is rotated in the clockwise direction.
Definition of $\cos$cos and $\sin$sin can extend beyond $0\le\theta\le\frac{\pi}{2}$0≤θ≤π2. |
As we move through different values of $\theta$θ the value of $\cos\theta$cosθ and $\sin\theta$sinθ move accordingly between $-1$−1 and $1$1. If we plot the values of $\cos\theta$cosθ and $\sin\theta$sinθ according to different values of theta on the unit circle, we get the following graphs:
$y=\cos\theta$y=cosθ |
$y=\sin\theta$y=sinθ |
As in the right-angled triangle definition, we still define $\tan\theta$tanθ as $\frac{\sin\theta}{\cos\theta}$sinθcosθ, which gives us the following graph:
$y=\tan\theta$y=tanθ |
By using the graph of $y=\cos x$y=cosx, what is the sign of $\cos\frac{23\pi}{12}$cos23π12?
Think: Using the graph of $y=\cos x$y=cosx, we can roughly estimate where the point $\left(\frac{23\pi}{12},\cos\frac{23\pi}{12}\right)$(23π12,cos23π12) lies and from this, determine the sign of $\cos\frac{23\pi}{12}$cos23π12.
Do: We plot the point on the graph of $y=\cos x$y=cosx below.
The point $\left(\frac{23\pi}{12},\cos\frac{23\pi}{12}\right)$(23π12,cos23π12) drawn on the graph of $y=\cos x$y=cosx. |
We can quickly observe that the height of the curve at this point is above the $x$x-axis, and observe that $\cos\frac{23\pi}{12}$cos23π12 is positive.
What quadrant does an angle with measure $\frac{23\pi}{12}$23π12 lie in?
Think: $\frac{23\pi}{12}$23π12 lies between $\frac{3\pi}{2}$3π2 and $2\pi$2π.
Do: An angle with a measure that lies between $\frac{3\pi}{2}$3π2 and $2\pi$2π is said to be in the fourth quadrant. So angle with measure $\frac{23\pi}{12}$23π12 lies in quadrant $IV$IV.
Reflect: The value of $\cos$cos is positive in the first and fourth quadrant and negative in the second and third quadrant. This holds true when we look at the graph of $y=\cos x$y=cosx as well.
Consider the equation $y=\sin x$y=sinx.
Using the fact that $\sin\frac{\pi}{6}=\frac{1}{2}$sinπ6=12, what is the value of $\sin\frac{5\pi}{6}$sin5π6?
Using the fact that $\sin\frac{\pi}{6}=\frac{1}{2}$sinπ6=12, what is the value of $\sin\frac{7\pi}{6}$sin7π6?
Using the fact that $\sin\frac{\pi}{6}=\frac{1}{2}$sinπ6=12, what is the value of $\sin\frac{11\pi}{6}$sin11π6?
Complete the table of values. Give your answers in exact form.
$x$x | $0$0 | $\frac{\pi}{6}$π6 | $\frac{\pi}{2}$π2 | $\frac{5\pi}{6}$5π6 | $\pi$π | $\frac{7\pi}{6}$7π6 | $\frac{3\pi}{2}$3π2 | $\frac{11\pi}{6}$11π6 | $2\pi$2π |
---|---|---|---|---|---|---|---|---|---|
$\sin x$sinx | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ |
Draw the graph of $y=\sin x$y=sinx.
Consider the equation $y=\cos x$y=cosx.
Complete the table of values. Give your answers in exact form.
$x$x | $0$0 | $\frac{\pi}{3}$π3 | $\frac{\pi}{2}$π2 | $\frac{2\pi}{3}$2π3 | $\pi$π | $\frac{4\pi}{3}$4π3 | $\frac{3\pi}{2}$3π2 | $\frac{5\pi}{3}$5π3 | $2\pi$2π |
---|---|---|---|---|---|---|---|---|---|
$\cos x$cosx | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ |
Draw the graph of $y=\cos x$y=cosx.
Given the unit circle, which of the following is true about the graph of $y=\tan x$y=tanx? Select all that apply.
The range of values of $y=\tan x$y=tanx is $-\infty
The graph of $y=\tan x$y=tanx repeats in regular intervals since the values of $\sin x$sinx and $\cos x$cosx repeat in regular intervals.
Since the radius of the circle is one unit, the value of $y=\tan x$y=tanx lies in the region $-1\le y\le1$−1≤y≤1.
The graph of $y=\tan x$y=tanx is defined for any measure of $x$x.
The range of values of $y=\tan x$y=tanx is $-\infty
The graph of $y=\tan x$y=tanx repeats in regular intervals since the values of $\sin x$sinx and $\cos x$cosx repeat in regular intervals.
Since the radius of the circle is one unit, the value of $y=\tan x$y=tanx lies in the region $-1\le y\le1$−1≤y≤1.
The graph of $y=\tan x$y=tanx is defined for any measure of $x$x.
Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions