Graphs of trigonometric functions

I'm making the assumption that you understand the unit circle and you already know how to plot one point of a function. For the three basic trigonometric functions the vertical axis is the ratio between the sides of the right triangle, while the horizontal axis is the angle in radians. To plot the inverse trigonometric functions we have the angle in the vertical axis, while the horizontal axis is the ratio.

Sine and cosine both have identical shapes. The sole difference is that sine and cosine differ in that [math]\displaystyle{ \sin(0) = 0 }[/math] and [math]\displaystyle{ \cos(0) = 1 }[/math], whereas [math]\displaystyle{ \sin(\pi/2) = 1 }[/math] and [math]\displaystyle{ \cos(\pi/2) = 0 }[/math]. Plotting the graph without attention to this fact can lead to mixing up one with the other and causing calculations to go wrong.

The sine and cosine waves

(they aren't perfect sines or cosines because they were hand drawn)

Why are they waves? You need to understand the unit circle and how to read it. First, angles as we learn in euclidean geometry, are always in between 0° and 360°. That is, from zero to doing a full circle turn. Any angle beyond that is just multiples of full turns or partial turns. With negative angles not meaning "an angle that is less than nothing" but just reflecting the fact that we can measure angles clockwise or anti-clockwise. This also explains that we are not required to plot the graph beyond one full turn, because anything beyond that is just repeating the same pattern.

At school is common to teach how to measure angles first, then the relationship between angles and ratios between sides of a right triangle. Lastly, trigonometry is combined with the definition of functions. By definition, sines and cosines map a measure, the angle, to a ratio between sides of the right triangle. Both angles and ratios are nothing more than numbers, irrational numbers most of the time. A trigonometric function is what relates those two numbers. Any number such as 1000 or 0.3 is treated as multiple turns, half turns, partial turns, etc along a circle for the purposes of trigonometric functions. That's why sines and cosines are waves that keep oscillating from 1 to -1 back and forth.

Choose a point anywhere on the perimeter of the unit circle. Use it to draw a right triangle with one of the vertexes being the center of the circle. Notice that whenever you move the point on the perimeter clockwise or anti-clockwise, the ratio between any two sides of the triangle keeps changing. This "proves" that the sine and cosine are not "zig-zag" graphs, but smooth waves.

Note: In case you know just a bit of numerical methods and wondered "Can I trace waves with parabolas?". Yes, that is possible. But here is the problem, one parabola is never a wave. What do you do? Break the wave into multiple segments, each one being the segment of a parabola. So you have a series of parabolas, half with upwards concavity, half with downwards concavity. Parabolas can never perfectly fit the curvature of a sine or cosine wave though.

An extra comment to add to the above. Sometimes people make the connection between the sine and cosine waves with the shape of the circle itself. Careful! The curvature of the sine and cosine waves are not that of half-circles! A parabola can be distorted such that it fits a half-circle but to understand that, one has to understand some concepts from analytical geometry. I'd theorize that this confusion comes from the fact that the unit circle can, erroneously, be seen as the graph of a function itself. The graph of a function can, at best, trace half of a circle. Most textbooks in calculus mention the "vertical line test" to differentiate graphs of functions from graphs of other equations which aren't functions.

The tangent

Let's move [math]\displaystyle{ p }[/math] to a position that is associated to the angle 45°. Just by doing this we get two values "for free", without having to calculate anything at all. [math]\displaystyle{ \theta^{\circ} = 45^{\circ} }[/math] because, in Euclidean geometry, the sum of all three angles of a triangle must be always 180°. The other value that we have is [math]\displaystyle{ a = r }[/math]. We know that we have a triangle with two angles equal to 45°. The only way for this to happen is to have two sides, excluding the hypotenuse, equal to each other. The line that is tangent to the circle at [math]\displaystyle{ p }[/math] is unique, there is only one and it happens to coincide with the side [math]\displaystyle{ a }[/math]. There another fact in the triangle that we have. The ratio [math]\displaystyle{ r/a = a/r = 1 }[/math] happens to be the length of the side [math]\displaystyle{ a }[/math]. Now, [math]\displaystyle{ r }[/math] is a constant and equal to one, it's much easier to calculate [math]\displaystyle{ a/1 = a }[/math] for every angle than the inverse of that.

Now move [math]\displaystyle{ p }[/math] anti-clockwise and close to the null angle (but keep the tangent perpendicular to the radius!). What happens with [math]\displaystyle{ a }[/math]? Its length becomes closer to zero. Move [math]\displaystyle{ p }[/math] in the other direction (but keep the tangent perpendicular to the radius!), clockwise and closer to the right angle. What happens with [math]\displaystyle{ a }[/math]? Its length stretches so much that if we reach the right angle, [math]\displaystyle{ a }[/math] goes to infinity. Hence, tangent for 90° doesn't exist. Another way to see this fact: what is the tangent to a perfectly plane surface? It would be a line that is parallel to the plane (we are talking about trajectories which are straight lines here, no parabolas), but in this case either the line is contained in the plane or doesn't touch it at all. Hence, this contradiction shows that tangent of 90° doesn't exist.

With sines and cosines we saw that the sides of the triangle never go beyond 1 unit, the triangle is always inside the unit circle. With tangent, however, the triangle has sides crossing the unit circle. Therefore, the graph won't be a wave limited by 1 and -1. For angles very close to 90° and 270° the graph is going to extend to almost vertical lines. For angles close to 0° or 180° it's going to be a curve, not a straight line, until it reaches zero. It's a function that associates the angle with the length of [math]\displaystyle{ a }[/math]. That's nothing new because sine and cosine are defined as relationships between angles and lengths of triangle's sides.

In case you wondered, there is another way to see the relationship between angles, sides and the tangent. Let's look at the same triangle that we had for sines and cosines and think on the ratio rise / run. For 45° it's pretty easy to see that rise = run. For 0° we have that [math]\displaystyle{ \sin(0) / \cos(0) = 0 }[/math]. For 90° we have that [math]\displaystyle{ \sin(\pi/2) = 1 }[/math] and [math]\displaystyle{ cos(\pi/2) = 0 }[/math] and we can't divide by zero. In other words, tangent is also a sin / cos ratio.