Radians (HSC SSCE Mathematics Advanced): Revision Notes

Radians

What is a radian?

A radian is a unit of angular measurement used frequently in mathematics. Unlike degrees, which divide a full circle into 360 equal parts, radians are based on the relationship between an angle and the arc it creates on a circle.

This definition leads to an important relationship: the radian measure of any angle can be calculated using the formula:

$\text{Radian measure} = \frac{\text{Arc length}}{\text{Radius of circle}}$

The relationship between degrees and radians

To understand how radians relate to the more familiar degree measure, consider a complete circle. The circumference of a circle with radius $r$ is given by $C = 2\pi r$.

For a unit circle (where the radius is 1), the circumference is $2\pi \times 1 = 2\pi$. Since the entire circumference represents a complete revolution of 360°, we can conclude that:

$2\pi \text{ radians} = 360°$

Converting between degrees and radians

Since $\pi$ radians equals 180°, we can create conversion formulas by using this as a conversion factor:

To convert degrees to radians: Multiply the degree measure by $\frac{\pi}{180°}$

To convert radians to degrees: Multiply the radian measure by $\frac{180°}{\pi}$

Common radian measures

It's helpful to memorise these commonly used angle measures in both degrees and radians:

Worked example: Converting degrees to radians

Worked example: Converting radians to degrees

Trigonometric ratios with radians

Trigonometric ratios ($\sin\theta$, $\cos\theta$, and $\tan\theta$) work with angles measured in radians just as they do with degrees. When working with radians, we use the unit circle where the radius equals 1.

The ASTC rule

The sign of each trigonometric ratio depends on which quadrant the angle is in. The ASTC rule is a helpful memory aid:

• Quadrant 1 ($0 < \theta < \frac{\pi}{2}$): All ratios are positive
• Quadrant 2 ($\frac{\pi}{2} < \theta < \pi$): Sine is positive, cosine and tangent are negative
• Quadrant 3 ($\pi < \theta < \frac{3\pi}{2}$): Tangent is positive, sine and cosine are negative
• Quadrant 4 ($\frac{3\pi}{2} < \theta < 2\pi$): Cosine is positive, sine and tangent are negative

Reference angles and the unit circle

When finding trigonometric ratios for angles in any quadrant, we use reference angles. A reference angle is the acute angle between the terminal side of the given angle and the horizontal axis.

The unit circle shows the symmetry of angles:

Worked example: Finding a trigonometric ratio using a reference angle

Exact values from special triangles

Exact values for multiples of $\frac{\pi}{6}$ and $\frac{\pi}{4}$ can be found using special triangles:

• The $30°-60°-90°$ triangle (which is $\frac{\pi}{6}-\frac{\pi}{3}-\frac{\pi}{2}$ in radians)
• The $45°-45°-90°$ triangle (which is $\frac{\pi}{4}-\frac{\pi}{4}-\frac{\pi}{2}$ in radians)

Worked example: Evaluating exact trigonometric values

Important note about calculators

Remember!