AOS1 Topic 6: Circular Functions (Radians Introduction)
Circular functions, also known as trigonometric functions, describe relationships between angles and ratios of side lengths in right-angled triangles, but they are also closely linked to circles. The most common circular functions are:
- Sine (\(\sin\))
- Cosine (\(\cos\))
- Tangent (\(\tan\))
- Cosecant (\(\csc\))
- Secant (\(\sec\))
- Cotangent (\(\cot\))
These functions are called circular because they can be defined in terms of the unit circle, a circle with radius 1, centered at the origin of the coordinate plane. As an angle rotates around the origin, the coordinates of the point where the terminal side of the angle intersects the unit circle define the sine and cosine of the angle:
\(\sin(\theta) = y\), \(\cos(\theta) = x\)
where \((x, y)\) are the coordinates of the point on the unit circle corresponding to the angle \(\theta\).
Understanding Radians
In trigonometry, angles are often measured in radians rather than degrees. Radians provide a more natural way to describe angles in terms of the unit circle. A radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.
One full revolution around a circle is \(360^\circ\), which is equivalent to \(2\pi\) radians. Therefore, we can relate degrees to radians using the following formulas:
\(180^\circ = \pi \, \text{radians}\)
\(1^\circ = \frac{\pi}{180} \, \text{radians}\)
\(1 \, \text{radian} = \frac{180^\circ}{\pi} \approx 57.2958^\circ\)
Why Use Radians?
Radians are preferred in mathematical analysis and calculus because they simplify the relationship between angles and circular functions. For example, in calculus, the derivative of \(\sin(x)\) with respect to \(x\) is \(\cos(x)\) only when \(x\) is measured in radians.
Converting Between Degrees and Radians
To convert from degrees to radians, use the following formula:
\(\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\)
For example, to convert \(90^\circ\) to radians:
\(90^\circ \times \frac{\pi}{180} = \frac{\pi}{2} \, \text{radians}\)
To convert from radians to degrees, use:
\(\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\)
For example, to convert \(\frac{\pi}{4}\) radians to degrees:
\(\frac{\pi}{4} \times \frac{180}{\pi} = 45^\circ\)
The following table displays the conversions of some special angles from degrees to radians.
| Angle in degrees | \(0^\circ\) |
\(30^\circ\) |
\(45^\circ\) |
\(60^\circ\) |
\(90^\circ\) |
\(180^\circ\) |
\(360^\circ\) |
|---|---|---|---|---|---|---|---|
| Angle in radians | \(0\) | \(\frac{\pi}{6}\) | \(\frac{\pi}{4}\) | \(\frac{\pi}{3}\) | \(\frac{\pi}{2}\) | \(\pi\) | \(2\pi\) |
Some values for the trigonometric functions are given in the following table.
| \(\theta\) | \(0\) | \(\frac{\pi}{6}\) | \(\frac{\pi}{4}\) | \(\frac{\pi}{3}\) | \(\frac{\pi}{2}\) |
|---|---|---|---|---|---|
| \(\sin(\theta)\) | \(0\) | \(\frac{1}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(1\) |
| \(\cos(\theta)\) | \(1\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{1}{2}\) | \(0\) |
| \(\tan(\theta)\) |
\(\frac{0}{1} = 0\) | \(\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}}\) | \(\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1\) | \(\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}\) | undefined |
Graphing Circular Functions
When circular functions are graphed, radians are typically used on the horizontal axis (representing the angle \(\theta\)) and the vertical axis represents the values of the function. For example, the graph of \(\sin(\theta)\) is a smooth wave that repeats every \(2\pi\) radians, or one full revolution around the unit circle.
The key properties of sine and cosine graphs include:
- Amplitude: The height of the wave (maximum value of 1 and minimum value of -1 for sine and cosine)
- Period: The length of one complete cycle, which is \(2\pi\) radians
- Phase Shift: Horizontal shifts in the graph based on the starting angle
These graphs demonstrate the periodic nature of circular functions, repeating every \(2\pi\) radians.
Applications of Circular Functions and Radians
Circular functions and radians are used in a wide range of applications, including:
- Physics, to model periodic motion such as pendulums and oscillations
- Engineering, in signal processing and electrical circuits
- Computer graphics, for rotations and transformations
- Astronomy, to measure the positions of planets and stars
In each of these fields, using radians instead of degrees allows for more accurate and efficient calculations.