Intro to sin(x), cos(x) and tan(x)

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.

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 counterclockwise 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

question 2

question 3