The graph of $\sin$ and $\cos$ is a characteristic smooth horizontal evenly undulating shape which oscillates between 1 and -1 with a period of $2\pi$, for reasons which should be pretty clear if you think about the [[Unit Circle Definitions of Trigonometric Functions|unit circle]], substituting $\theta$ for $x$. ![[Sinusoidal Functions 1.png]] [Geogebra](https://www.geogebra.org/calculator/qz274bda) It has the following [[Amplitude, Period and Midline of Trigonometric Expressions|notable characteristics]]: 1. the graphs of $y=\sin(x)$ and $y=-\sin(x)$ and similarly $y=\cos(x)$ and $y=-\cos(x)$ are vertically symmetrical about a _midline_. 2. A vertical line through any peak or trough is a line of horizontal symmetry. 3. The the maximum distance a given sinusoidal function travels from the midline is known as the _amplitude_. This is half the distance between the peaks (maxima) and the troughs (minima). 4. It repeats horizontally after a given _period_. Thus $\sin$ and $\cos$ are not one-to-one mappings and would only be [[Inverse Trigonometric Functions|invertible]] over a restricted domain. For the basic sinusoidal functions $y=\sin x$ and $y=\cos x$ depicted above, it can be seen that: 1. The midline is $y=0$ 2. The amplitude is $1$ (maxima are at $x=1$, minima are at $x=-1$). 3. The period is $2\pi$, corresponding to a full rotation of the circle. This is assuming we are measuring in [[Radians]]. If we were using degrees, the period would be 360$\textdegree$ for obvious reasons. The general form of equations of graphs of these two functions could be written as follows: $ \begin{align*} y-k &= a \sin\left(m(x-h)\right)\\ y-k &= a \cos\left(m(x-h)\right) \tag{1}\\ \end{align*} $ Where: - $|a|$ is the _amplitude_, with the sign parameter in front flipping the graph about the x-axis if negative. - $\tau = \frac{2\pi}{|m|}$ is the _period_. If $m<0$ it is conventional to rewrite using [[Geometric Construction of Trigonometric Identities from the Unit Circle|identities]] such that $m > 0$. This means that any mirroring of the wave form about the $x$ axis comes from the $a$ parameter and they don't interact. It's worth experimenting in a graphing tool with doing or not doing this to make sure you understand degenerate expressions you might come across. Note that since $\frac{2\pi}{|m|}$ is the period, when modeling word problems etc values for $m$ will frequently be something like $\frac{2\pi}{365}$ for a function that has an annual period or $\frac{2\pi}{24}$ for something where a cycle lasts 1 day. - $f = \frac{1}{\tau} = \frac{|m|}{2\pi}$ is the _frequency_, measured in Hz (cycles per second). - $h$ is a _phase shift_ in the horizontal direction, such that as $h$ increases, $x-h$ gets smaller and the graph shifts to the _right_. A phase shift of $\pi$ makes the graph completely out of phase, and thus the graphs of $y=\sin(x+\frac{\pi}{2})$ and $y=\cos(x)$ are the same (which we already know from [[Trigonometric Identities from Rotation|rotation identities]]), as are the graphs of $y=\sin(x), y=-\sin(-x), y=-\sin(x+\pi),$ and $y=\cos(\frac{\pi}{2}-x)$ etc. It is more conventionally written as $\phi = -mh$, which is called the _phase angle_. - $k$ is a _vertical shift_ such that as $k$ increases, $y-k$ gets smaller and the midline shifts _up_. It can be seen that the $k$ and $h$ parameters with the equation in this form behave similarly to the corresponding parameters in the completed square form of a [[Quadratic Functions and Their Graphs|quadratic]]. This form is extremely helpful for the purposes of modelling things with sinusoidal functions eg in word problems. ## Standard form An important conventional form that you'll see in things like the [[Harmonic Oscillator|harmonic oscillator]] is as follows $ x(t) = a \cos(\omega t + \phi) + x_{eq}$ Where: - $a$ is the amplitude - $\omega$ is called the _angular frequency_ and determines the periodicity of the oscillator - $\phi$ is called the _phase angle_ and determines the horizontal translation, and - $x_{eq}$ is the _equilibrium position_, meaning the natural length of the spring, rest position of a pendulum etc. ## Varying parameters one by one ### Amplitude ($a$) Varying the $a$ parameter varies the heights of the peaks and troughs of the function. From a translation point of view this is dilation in the $y$ direction. ![[Sinusoidal Functions 4 amplitude.png]] Flipping the sign inverts the wave form,mirroring the original function about the $x$ axis: ![[Sinusoidal Functions 4 amplitude 2.png]] ### Period ($\frac{2\pi}{|m|}$) Altering the $m$ parameter affects the period. From a translation point of view this is dilation in the $x$ direction. If $m<0$, it's conventional to use trig identities to rewrite the expression to an equivalent expression where this is not the case. If not, $m$ will also invert the graph of $\sin$, but not $\cos$ which you may recall from the [[Unit Circle Definitions of Trigonometric Functions|unit circle]]. ![[Sinusoidal Functions 4 period.png]] ### Phase shift($h$) The $h$ parameter causes a horizontal translation. In the form given above, as $h$ gets larger, $x-h$ gets smaller so the the graph shifts to the _right_ and vice versa for as $h$ gets smaller the graph shifts to the _left_. if written as $x+h$ then clearly the opposite is true. ![[Sinusoidal Functions 4 phase shift.png]] ### Vertical shift($k$) The $k$ parameter causes a vertical shift. As written, as $k$ gets larger, $y-k$ gets smaller and the graph shifts _up_ and vice versa as $k$ gets smaller. ![[Sinusoidal Functions 4 vertical shift.png]] ### Worked example 1: Graph $y=\frac{1}{2} \cos(\frac{1}{3}x + \frac{2\pi}{3}) -2$ Say we want to graph $y=\frac{1}{2} cos(\frac{1}{3}x + \frac{2\pi}{3}) -2$. First we rewrite into the standard form: $ \begin{align*} y&=\frac{1}{2} \cos(\frac{1}{3}x + \frac{2\pi}{3}) -2 \\ \Rightarrow y+2&=\frac{1}{2} \cos\left(\frac{1}{3}(x + 2\pi)\right)\\ \end{align*} $ we can see from the above that the amplitude should be $\frac{1}{2}$. The midline is $-2$, the the period is $\frac{2\pi}{\frac{1}{3}} = 6\pi$ and the phase shift is $2\pi$ (or a third of a cycle) to the _left_. ![[Sinusoidal Functions 2.png]] [geogebra](https://www.geogebra.org/calculator/mghdhwaw) To test out the impact of various parameters, why not use the geogebra link above to get that graph and try different values? ## Deriving graphs of sinusoidal functions by translation To derive the graph of an arbitrary sinusoidal function by translation from the graph of $y=\sin x$ or $y = \cos x$, the process is as follows. 1. rewrite into the standard form above, including rewriting to get a positive $m$ as necessary. 2. If the $a<0$, mirror about the x-axis 3. dilate vertically by the amplitude($|a|$) 4. dilate horizontally by $\frac{1}{|m|}$ 5. translate vertically by $k$ 6. translate horizontally by $h$ The order of steps 3 & 4 and steps 5 and 6 can be reversed without ill effects. So for the example above, the steps to derive the graph from the graph of $y=\cos(x)$ would be 1. scale vertically by $\frac{1}{2}$ 2. scale horizontally by $3$ 3. shift downwards by $2$ 4. shift to the left by $2\pi$ ## Range and Image set of sinusoidal functions As we have seen, the range of a sinusoidal function is generally $\mathbf{R}$ unless a specific restriction is added, and the image set of $y=\sin x$ and $y=\cos x$ is $[-1,1]$. The image set of any arbitrary sinusoidal function in the form at (1) is going to be $[k-a,k+a]$. ## Limits and continuity of sinusoidal functions Its intuitively obvious that these functions are continuous but in the interests of completeness we need to prove this. [[Limits and continuity of sinusoidal functions|TODO]] ## Inverse of a sinusoidal function We know that the [[Inverse Trigonometric Functions]] exist and that, since $\sin$ and $\cos$ are not one-to-one mappings, that we would need to restrict the domain for an inverse to exist. The process to go from such a restricted version of a general sinusoidal function to its inverse will be mapped here. TODO ## Composite sinusoidal functions Periodic waves of arbitrary complexity can be modelled by adding together scaled sinusoidal functions. The process of finding the right functions to model a particular wave is known as [Fourier Analysis](https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/Book%3A_AC_Electrical_Circuit_Analysis%3A_A_Practical_Approach_(Fiore)/01%3A_Fundamentals/1.3%3A_Basic_Fourier_Analysis). ![[Composite Sinusoidal Function.png]]