Defining Circular Functions: Sine, Cosine, and Tangent (VCE SSCE Mathematical Methods): Revision Notes

Defining Circular Functions: Sine, Cosine, and Tangent

Introduction to the unit circle

In circular functions, we work with a unit circle, which is a circle with radius 1 centred at the origin of a coordinate plane. Each point on this circle corresponds to an angle $\theta$ measured from the positive $x$-axis in an anticlockwise direction.

The point on the unit circle corresponding to angle $\theta$ is written as $P(\theta)$. The position of this point is completely determined by the angle, which means both the $x$-coordinate and $y$-coordinate depend on $\theta$.

Defining sine and cosine

The coordinates of point $P(\theta)$ on the unit circle give us our definitions of sine and cosine:

The $x$-coordinate of $P(\theta)$ is given by:

$$x = \cos \theta, \text{ for } \theta \in \mathbb{R}$$

The $y$-coordinate of $P(\theta)$ is given by:

$$y = \sin \theta, \text{ for } \theta \in \mathbb{R}$$

This means we can write the coordinates of any point on the unit circle as:

$$P(\theta) = (\cos \theta, \sin \theta)$$

This fundamental relationship shows that cosine gives the horizontal position and sine gives the vertical position of any point on the unit circle.

Periodicity of sine and cosine

An important property of circular functions is their periodicity. When you add $2\pi$ to any angle, you complete one full rotation around the circle and return to the same point. This means:

$$\cos(2\pi + \theta) = \cos \theta$$ $$\sin(2\pi + \theta) = \sin \theta$$

More generally, for any integer $k$:

$$\sin(2k\pi + \theta) = \sin \theta$$ $$\cos(2k\pi + \theta) = \cos \theta$$

Defining tangent

The tangent function is defined differently. If we draw a tangent line to the unit circle at point $A(1, 0)$, and extend the line from the origin through $P(\theta)$ until it intersects this tangent line at point $C$, then the y-coordinate of C is called tangent $\theta$.

Using similar triangles $OPD$ and $OCA$, we can show that:

$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$

Domain of tangent

The tangent function is undefined when cosine equals zero. This occurs when:

$$\theta = \pm\frac{\pi}{2}, \pm\frac{3\pi}{2}, \pm\frac{5\pi}{2}, \ldots$$

Periodicity of tangent

Adding $\pi$ to the angle doesn't change the line $OP$, so tangent has a period of π:

$$\tan(\pi + \theta) = \tan \theta$$

For any integer $k$:

$$\tan(k\pi + \theta) = \tan \theta$$

Exact values of circular functions

While calculators can find approximate values for most angles, there are special angles where we can determine exact values using geometry.

Exact values for $0$ and $\frac{\pi}{2}$ (0° and 90°)

From the unit circle, we can see directly:

• $\sin 0° = 0$, $\cos 0° = 1$, $\tan 0° = 0$

• $\sin 90° = 1$, $\cos 90° = 0$, $\tan 90°$ is undefined

  

Exact values for $\frac{\pi}{6}$ and $\frac{\pi}{3}$ (30° and 60°)

These values come from an equilateral triangle with side length 2 units. When we draw a perpendicular from one vertex to the opposite side, we create two right-angled triangles. Using Pythagoras' theorem, the height is $\sqrt{3}$.

From this triangle:

• $\sin 30° = \frac{1}{2}$, $\cos 30° = \frac{\sqrt{3}}{2}$, $\tan 30° = \frac{1}{\sqrt{3}}$

• $\sin 60° = \frac{\sqrt{3}}{2}$, $\cos 60° = \frac{1}{2}$, $\tan 60° = \sqrt{3}$

  

Exact values for $\frac{\pi}{4}$ (45°)

These values come from an isosceles right-angled triangle with legs of length $1$ unit. The hypotenuse has length $\sqrt{2}$.

From this triangle:

• $\sin 45° = \frac{1}{\sqrt{2}}$, $\cos 45° = \frac{1}{\sqrt{2}}$, $\tan 45° = 1$

Table of exact values

Symmetry properties of circular functions

The coordinate axes divide the unit circle into four quadrants, numbered anticlockwise from the positive $x$-axis:

Using symmetry, we can relate the values of circular functions for angles in different quadrants to a reference angle in the first quadrant.

Key symmetry relationships

If $P(\theta) = (\cos \theta, \sin \theta) = (a, b)$ is in Quadrant 1, then:

Quadrant 2 ($\pi - \theta$):

• $\sin(\pi - \theta) = \sin \theta$
• $\cos(\pi - \theta) = -\cos \theta$
• $\tan(\pi - \theta) = -\tan \theta$

Quadrant 3 ($\pi + \theta$):

• $\sin(\pi + \theta) = -\sin \theta$
• $\cos(\pi + \theta) = -\cos \theta$
• $\tan(\pi + \theta) = \tan \theta$

Quadrant 4 ($2\pi - \theta$):

• $\sin(2\pi - \theta) = -\sin \theta$
• $\cos(2\pi - \theta) = \cos \theta$
• $\tan(2\pi - \theta) = -\tan \theta$

Signs of circular functions: the CAST rule

The symmetry properties tell us which trigonometric functions are positive in each quadrant:

• Quadrant 1: All functions are positive (A)
• Quadrant 2: Sine is positive (S)
• Quadrant 3: Tangent is positive (T)
• Quadrant 4: Cosine is positive (C)

Negative angles

When we use negative angles (measured clockwise from the positive x-axis), the following relationships hold:

By symmetry across the $x$-axis:

$$\sin(-\theta) = -\sin \theta$$ $$\cos(-\theta) = \cos \theta$$ $$\tan(-\theta) = \frac{-\sin \theta}{\cos \theta} = -\tan \theta$$

This tells us:

• Sine is an odd function
• Cosine is an even function
• Tangent is an odd function

Worked examples