Unit Circle (HSC SSCE Mathematics Advanced): Revision Notes

Unit Circle

What is the unit circle?

The unit circle is a circle with radius $1$ that is centred at the origin of the Cartesian coordinate plane. This special circle allows us to extend the definitions of trigonometric ratios beyond the constraints of right-angled triangles.

When we position a right-angled triangle at the origin of a coordinate plane with a hypotenuse of length $1$, the trigonometric ratios can be expressed directly in terms of the coordinates $x$ and $y$.

Trigonometric ratios on the unit circle

For any right-angled triangle positioned on the unit circle, with the hypotenuse having length $1$, the three basic trigonometric ratios are:

$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{y}{1} = y$

$\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{x}{1} = x$

$\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{y}{x}$

When we examine any right-angled triangle with hypotenuse of length $1$, the endpoint of the hypotenuse lies on the unit circle. We measure angles from the positive $x$-axis to the radius that extends to any point on the circle.

Coordinates on the unit circle

The angle $\theta$ is formed by rotating anticlockwise from the positive $x$-axis. Since $\cos \theta = x$ and $\sin \theta = y$, we can represent the coordinates of any point on the unit circle as:

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

The gradient of the line segment from the origin to point $P$ can be expressed using the ratio $\frac{y}{x}$:

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

Therefore, $m = \tan \theta$, where $m$ represents the gradient.

Worked example: Finding exact trigonometric values

Signs of trigonometric ratios

The unit circle demonstrates that $\sin \theta$, $\cos \theta$, and $\tan \theta$ are defined for angles extending beyond what can be represented in a right-angled triangle. The sign (positive or negative) of each ratio depends on which quadrant the angle terminates in.

Signs in each quadrant

Second quadrant:

• $y$ is positive, so $\sin \theta$ is positive
• $x$ is negative, so $\cos \theta$ is negative
• The gradient is negative, so $\tan \theta$ is negative

Third quadrant:

• $y$ is negative, so $\sin \theta$ is negative
• $x$ is negative, so $\cos \theta$ is negative
• The gradient is positive, so $\tan \theta$ is positive

Fourth quadrant:

• $y$ is negative, so $\sin \theta$ is negative
• $x$ is positive, so $\cos \theta$ is positive
• The gradient is negative, so $\tan \theta$ is negative

The ASTC rule

A helpful mnemonic for remembering which trigonometric functions are positive in each quadrant is "All Stations To Central" (ASTC).

Worked example: Determining the sign of a ratio

Trigonometric ratios for any angle magnitude

The unit circle extends the definitions of trigonometric ratios to include angles of any magnitude. This means we can evaluate $\sin \theta$, $\cos \theta$, and $\tan \theta$ for positive angles, negative angles, and angles beyond $360°$.

For any point $P(x, y)$ on the unit circle (where the radius $r = 1$), rotated by angle $\theta$ anticlockwise from the positive $x$-axis:

• $\cos \theta = x$
• $\sin \theta = y$
• $\tan \theta = \frac{y}{x}$ (undefined when $x = 0$)

Working with angles of any magnitude

We can represent angles of any magnitude by adding or subtracting multiples of $360°$:

$\theta \pm 360° \times n$

where $n$ is an integer.

For negative angles, we rotate clockwise from the positive $x$-axis. For example, $\theta = -30°$ is equivalent to rotating $360° - 30° = 330°$ anticlockwise.

When the angle equals $360°$, it occupies the same position as $0°$, so its related angle is $0°$.

Worked example: Evaluating ratios for negative angles

Worked example: Finding coordinates for large angles

Remember!