AOS1 Topic 6: Circular Functions (COS)

Circular functions, also known as trigonometric functions , are fundamental in the study of angles and periodic phenomena. One of the most important circular functions is the cosine function , abbreviated as $\cos$. It is widely used in mathematics, physics, engineering, and other fields to model periodic behavior such as waves, oscillations, and circular motion.



Definition of the Cosine Function


1. In Right-Angled Triangles


In a right-angled triangle, the cosine of an angle θthetaθ is defined as the ratio of the length of the adjacent side to the length of the hypotenuse :


\[cos( \theta)= \frac{ \text{Adjacent}}{ \text{Hypotenuse}}\]


2. On the Unit Circle


For an angle $\theta$ in the unit circle (a circle with radius 1 centered at the origin), the cosine of $\theta$ represents the horizontal coordinate of the point on the unit circle corresponding to $\theta$.


\[\cos(\theta) = x\]


where $(x,y)$ are the coordinates of the point on the unit circle corresponding to $\theta$.



Cosine on the Unit Circle

The unit circle provides a powerful way to define the cosine function for any angle, not just those between $0$ and $90$ degrees (or $0$ to $\frac{\pi}{2}$​ radians).


\(\theta\) (Radians)\(\cos(\theta)\)\(\theta\) (Degrees)


\(0\)\(1\)\(0^\circ\)
\(\frac{\pi}{6}\)\(\frac{\sqrt{3}}{2}\)\(30^\circ\)
\(\frac{\pi}{4}\)\(\frac{\sqrt{2}}{2}\)\(45^\circ\)
\(\frac{\pi}{3}\)\(\frac{1}{2}\)\(60^\circ\)
\(\frac{\pi}{2}\)\(0\)\(90^\circ\)
\(\pi\)\(-1\)\(180^\circ\)
\(\frac{3\pi}{2}\)\(0\)\(270^\circ\)
\(2\pi\)\(1\)\(360^\circ\)


Periodicity of the Cosine Function


The cosine function is periodic , meaning it repeats its values at regular intervals. Specifically, $\cos(\theta)$ has a period of $2\pi$ radian. So $ 4\pi$ is the same as $2\pi$:


\[\cos(\theta + 2\pi) = \cos(\theta)\]


This periodicity is essential for modeling waves and oscillations.



Graph of the Cosine Function


The graph of y = cos(θ) is a wave-like curve that oscillates between $1$ and $-1$, repeating every $2\pi$ radians.


Key characteristics of the cosine graph include:


Amplitude: The height of the wave from the centerline ($x = 0$) to the peak. For $y = \cos(\theta)$, the amplitude is $1$.


Period: The distance along the x-axis for one complete cycle of the wave. For cosine, the period is $2\pi$.


Phase Shift: The horizontal shift along the x-axis. The standard cosine function starts at (0,1), but phase shifts occur when the function is modified, for example $\cos( \theta − \phi)$, where $\phi$ is the phase shift. (Explained more in Transformations topic)


Vertical Shift: If a constant is added or subtracted, the entire graph moves up or down. (Explained more in Transformations topic)



Properties of the Cosine Function


Even Function: Cosine is an even function, meaning that $\cos(− \theta) = \cos( \theta)$. This symmetry reflects across the y-axis.


Range: The values of the cosine function are always between $-1$ and $1$, i.e., $−1 \leq \cos( \theta) \leq 1$.


Periodicity: As mentioned earlier, the cosine function repeats every $2\pi$ radians. This periodic nature is essential for modeling cyclic phenomena like sound waves or seasonal patterns.



Applications of the Cosine Function


The cosine function is applied in numerous fields, including:


Physics: To describe harmonic motion, waveforms, and alternating current (AC) circuits.


Engineering: For analyzing signal processing, vibrations, and structures involving oscillations.


Geometry and Trigonometry: In the calculation of distances, angles, and for solving triangles.


Astronomy: For modeling planetary orbits, which are often elliptical but can be approximated using circular functions.


Music and Acoustics: Cosine functions are used to describe sound waves and harmonics.



Summary: Cosine Function ($\cos(\theta)$)


Definition:


In right-angled triangles: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$​.


On the unit circle: $\cos(\theta)$ is the x-coordinate of the point corresponding to angle $\theta$.



Unit Circle Characteristics:


Cosine values range between $-1$ and $1$.


Key angles have specific cosine values (e.g., $\cos(0) = 1$, $\cos(\pi) = -1$).



Graphical Properties:


Amplitude: 1


Period: $2\pi$ radians


Phase Shift: Horizontal shifts based on function transformations.


Vertical Shift: Upward or downward shifts based on constants added or subtracted.



Mathematical Properties:


Even Function: $\cos(-\theta) = \cos(\theta)$.


Range: $-1 \leq \cos(\theta) \leq 1$.


Example 1

Finding exact value of trigonometric functions
Find the exact value of:
\(\cos(-585^\circ)\)

Solution:
\(\cos(-585^\circ) = \cos 585^\circ = \cos(585^\circ - 360^\circ) = \cos 225^\circ = -\cos 45^\circ = -\frac{1}{\sqrt{2}}\)

Example 2

Find the exact value of: \( \cos\left(-\frac{45\pi}{6}\right) \).

Solution:
\( \cos\left(-\frac{45\pi}{6}\right) = \cos\left(-\frac{15}{2} \times \pi\right) = \cos\left(\frac{\pi}{2}\right) = 0 \)

Example 3

Transformations of the graphs of sine and cosine
Sketch the graph of the function
\( h: [0, 2\pi] \rightarrow \mathbb{R} \)
\( h(x) = 3 \cos\left(2x + \frac{\pi}{3}\right) + 1 \)

Solution
We can write \( h(x) = 3 \cos\left(2x + \frac{\pi}{6}\right) + 1 \).
The graph of \( y = h(x) \) is obtained from the graph of \( y = \cos x \) by:
  • a dilation of factor \(\frac{1}{2}\) from the y-axis
  • a dilation of factor 3 from the x-axis
  • a translation of \(\frac{\pi}{6}\) units in the negative direction of the x-axis
  • a translation of 1 unit in the positive direction of the y-axis.
    • First apply the two dilations to the graph of \( y = \cos x \).

    Translation
    Next apply the translation \(\frac{\pi}{6}\) units in the negative direction of the x-axis.

    Final Translation
    Apply the final translation and restrict the graph to the required domain.

    Exercise 1

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