NZ Level 8 (NZC) Level 3 (NCEA) [In development]
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}$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

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

##### question 2

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

1. Complete the table of values. Give your answers in exact form.

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

##### 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