NZ Level 8 (NZC) Level 3 (NCEA) [In development]
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Intro to sin(x), cos(x) and tan(x)
Lesson

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.

Right-angled triangle definition

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}$0θπ2. 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}$0θπ2.

 

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

 

Worked example

example 1

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.

example 2

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.

Practice questions

question 1

Consider the equation $y=\sin x$y=sinx.

  1. Using the fact that $\sin\frac{\pi}{6}=\frac{1}{2}$sinπ6=12, what is the value of $\sin\frac{5\pi}{6}$sin5π6?

  2. Using the fact that $\sin\frac{\pi}{6}=\frac{1}{2}$sinπ6=12, what is the value of $\sin\frac{7\pi}{6}$sin7π6?

  3. Using the fact that $\sin\frac{\pi}{6}=\frac{1}{2}$sinπ6=12, what is the value of $\sin\frac{11\pi}{6}$sin11π6?

  4. 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 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
  5. Draw the graph of $y=\sin x$y=sinx.

    Loading Graph...

question 2

Consider the equation $y=\cos x$y=cosx.

  1. 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 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
  2. Draw the graph of $y=\cos x$y=cosx.

    Loading Graph...

question 3

Given the unit circle, which of the following is true about the graph of $y=\tan x$y=tanx? Select all that apply.

  1. The range of values of $y=\tan x$y=tanx is $-\infty<y<.

    A

    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.

    B

    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$1y1.

    C

    The graph of $y=\tan x$y=tanx is defined for any measure of $x$x.

    D

    The range of values of $y=\tan x$y=tanx is $-\infty<y<.

    A

    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.

    B

    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$1y1.

    C

    The graph of $y=\tan x$y=tanx is defined for any measure of $x$x.

    D

Outcomes

M8-2

Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions

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