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. They are also closely linked to circles, particularly the unit circle. The most common circular functions are:
- Sine (\(\sin\))
- Cosine (\(\cos\))
- Tangent (\(\tan\))
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.
Definition of a Radian
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.
In other words, a radian is the angle at which the arc length is the radius. This angle is the same for all circles no matter how big or small the radius is.
Relationship Between Degrees and Radians
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
From Degrees to 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:
\[90^\circ \times \frac{\pi}{180} = \frac{\pi}{2} \, \text{radians}\]
From Radians to Degrees
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:
\(\frac{\pi}{4} \times \frac{180}{\pi} = 45^\circ\)
Created with GeoGebra ® , by jimfay55, Link
Conversion Table for Special Angles.
| Angle in degrees | \(0^\circ\) | \(30^\circ\) | \(45^\circ\) | \(60^\circ\) | \(90^\circ\) | \(180^\circ\) | \(360^\circ\) |
| Angle in radians | \(0\) | \(\frac{\pi}{6}\) | \(\frac{\pi}{4}\) | \(\frac{\pi}{3}\) | \(\frac{\pi}{2}\) | \(\pi\) | \(2\pi\) |
Values of Trigonometric Functions for Special Angles
The following table displays the values of sine, cosine, and tangent for some special angles:
| \(\theta\) | \(0\) | \(\frac{\pi}{6}\) | \(\frac{\pi}{4}\) | \(\frac{\pi}{3}\) | \(\frac{\pi}{2}\) |
| \(\sin(\theta)\) | \(0\) | \(\frac{1}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(1\) |
| \(\cos(\theta)\) | \(1\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{1}{2}\) | \(0\) |
| \(\tan(\theta)\) | \(0\) | \(\frac{1}{\sqrt{3}}\) | \(1\) | \(\sqrt{3}\) | 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.
Key Properties of Sine and Cosine Graphs
- Amplitude: The height of the wave. For sine and cosine functions, the amplitude is $1$ (maximum value of $1$ and minimum value of $-1$).
- Period: The length of one complete cycle, which is $2\pi$ radians.
- Phase Shift: Horizontal shifts in the graph based on the starting angle.
- Vertical Shift: Vertical shifts in the graph based on the function's equation
These graphs demonstrate the periodic nature of circular functions, repeating every \(2\pi\) radians.
Created with GeoGebra ® , by Gaetano Di Caprio, Link
Practise Memorising the Unit Circle on this website
https://www.mathwarehouse.com/unit-circle/unit-circle-game.php
Applications of Circular Functions and Radians
Circular functions and radians are used in a wide range of applications, including:
Physics: Modeling periodic motion such as pendulums, oscillations, and waves.
Engineering: Signal processing, electrical circuits, and mechanical vibrations.
Computer Graphics: Rotations, transformations, and animations.
Astronomy: Measuring the positions and movements of planets and stars.
Navigation: Calculating bearings and distances using trigonometry.
In each of these fields, using radians instead of degrees allows for more accurate and efficient calculations, especially in calculus and advanced mathematical modeling.
Common Misconceptions
Misconception 1: Radians and Degrees Are Interchangeable in All Calculations
Clarification: While radians and degrees can be converted into each other, certain formulas and calculations in calculus and trigonometry require angles to be in radians for accuracy. For example, the derivative of $\sin(x)$ is $\cos(x)$ only when $x$ is in radians.
Summary: Circular Functions and Radians
Circular Functions:
Sine ($\sin$), Cosine ($\cos$), Tangent ($\tan$) describe relationships between angles and side ratios in right-angled triangles.
Defined using the unit circle , where $\sin(\theta) = y$ and $\cos(\theta) = x$ for a point $(x, y)$ on the unit circle.
Radians:
Radians provide a natural way to measure angles based on the unit circle.
Conversion Formulas:
\[180^\circ = \pi \text{ radians}\]
\[1^\circ = \frac{\pi}{180} \text{ radians}\]
\[1 \text{ radian} =180^\circ \pi \approx 57.2958^\circ\]
Trigonometric Function Values:
Special angles have known sine, cosine, and tangent values, which are essential for solving trigonometric equations and modeling periodic phenomena.