Trigonometry

The easiest thing to keep in mind is thinking about a circle with radius $1$. Choose some angle $\alpha$ (alpha). Draw a line which has that angle counter-clockwise from the $x$-axis through the origin $\mathbf{O}$. The line meets the circle in a point $\mathbf{P}$. Draw a straight line through $\mathbf{P}$ parallel to the $y$-axis, which will intersect the $x$-axis in a point $\mathbf{Q}$. You now have a right triangle $\Delta \mathbf{O}\mathbf{Q}\mathbf{P}$, where the right angle occurs at the point $\mathbf{Q}$.

The sine $\sin$ is just the $y$-segment from $\mathbf{Q}$ to $\mathbf{P}$. It is positive, if the line goes up and negative if it goes down.

The cosine $\cos$ is just the $x$-segment from $\mathbf{O}$ to $\mathbf{Q}$. It is positive, if the line goes right and negative if it goes left.

The segment from $\mathbf{O}$ to $\mathbf{P}$ is called the hypotenuse and its length is equal to the circle radius, which is $1$ in our case.

Below you can see a visualization of the sine and cosine. You can move around the point on the circle. This changes the angle $\alpha$, showing you how it relates to the different values of $\sin$ and $\cos$.

In the visualization you will also see to additional values: The tangent ($\tan$) and the cotangent ($\cot$). These are not as important as the sine and cosine, especially the cotangent, but the tangent is encountered pretty often. In the visualization you can see that its value is the height (positive or negative) of where the line through the angle vector intersects the y-axis at $x=1$. They can be defined as follows:

$$\begin{aligned} \tan(\alpha) &= \frac{\sin(\alpha)}{\cos(\alpha)} \\ \cot(\alpha) &= \frac{\cos(\alpha)}{\sin(\alpha)} \end{aligned}$$

If the circle didn't have radius $r=1$, but instead just some $r$, the whole triangle would look exactly the same, but scaled. So the side lengths would become $r\sin(\alpha)$ and $r\cos(\alpha)$.

From the Pythagorean theorem, we know that the sum of the squared shorter sides of a right triangle is equal to the squared length of the longer hypotenuse. Since we have a right triangle and used a unit circle with $r=1$, we have the important property:

$$\begin{aligned} \sin^2(\alpha) + \cos^2(\alpha) = 1 \end{aligned}$$

This comes up quite often, so it is important to memorize.

Another important thing to remember, is that in general, angles are specified in radians, not degrees, so it is good to get familiar with it. Radians measure the length of an arc segment on the unit circle, that is covered by an angle. If you can't remember how to switch between them, here is a simple way to remember it. Degrees and radians are proportional to each other. The circumference of the unit circle is $2\pi$. A circle has $360^\circ$. Dividing both angles by their respective full circle will yield the same ratio (a half circle is a half circle). With that we can relate an angle in degree $\alpha$ and in radians $b$ as:

$$\begin{aligned} \frac{\alpha}{360^\circ} &= \frac{b}{2\pi} \\ \alpha &= 360^\circ\frac{b}{2\pi} \\ &= 180^\circ \frac{b}{\pi} \\ b &= \frac{2\pi\alpha}{360^\circ}\\ &= \frac{\pi\alpha}{180^\circ} \end{aligned}$$

From the visualization above, you can deduce a number of special angle-number pairs, which are good to know:

$$\begin{aligned} \sin(0^\circ) &= \sin(0) = 0\\ \cos(0^\circ) &= \cos(0) = 1\\ \sin(90^\circ) &= \sin(\frac{\pi}{2}) = 1\\ \cos(90^\circ) &= \cos(\frac{\pi}{2}) = 0 \\ \sin(180^\circ) &= \sin(\pi) = 0\\ \cos(180^\circ) &= \cos(\pi) = -1 \\ \sin(270^\circ) &= \sin(\frac{3\pi}{2}) = -1\\ \cos(270^\circ) &= \cos(\frac{3\pi}{2}) = 0 \\ \end{aligned}$$

The diagonal angles ($45^\circ, 135^\circ, 225^\circ, 315^\circ$) also have a trick to remember them. In those cases the right triangle is symmetric and so the $\sin$ and $\cos$ sides must have the same length, let's call it $l$. We have $\sin^2 + \cos^2 = 1 \Rightarrow l^2 + l^2 = 2l^2 = 1 \Rightarrow l = \sqrt(\frac{1}{2})$. You just have to add a negative sign to this length, whenever the sine goes downward or the cosine left.

With this knowledge, you can already solve a lot of problems without computing any values. Just for completeness sake, let us look at the plots of both functions.

Having this graph roughly in the back of your mind, you can make some decent guesses at values for quick estimations.

You might also notice, that both graphs look very similar. That is because they are! Sine and cosine are actually the same graph, just offset a bit.

Next up are vectors!