For the earlier notes, see Wednesday 20 November
The notes below are still incomplete – and I want to add illustrations as well. But the ancient computer I am using these days connects to the internet slowly if at all, so I am already late posting this. I will add to it and update when I can.
Important versions of the two special triangles: when the hypotenuse is 1.
You can see these two right triangles on this blog post from squarerootofnegativeoneteachmath (which also shows how they are used in the unit circle definitions of the trig functions, which we will discuss next time).
Embedding a right triangle in the coordinate plane:
We pick out one of the acute angles in the right triangle: let’s call it angle $\theta$. We say that the right triangle is in standard position in the coordinate plane if the vertex of angle $\theta$ is at the origin, the side adjacent to angle $\theta$ is on the positive x-axis, and the hypotenuse extends into the first quadrant. This desmos graph shows a right triangle in standard position. Notice that the coordinates of the top right vertex are $(a, b)$ where $a$ and $b$ are the lengths of the two legs of the right triangle.
We are going to use this embedding to extend the definitions of the trig functions to angles which are not acute angles, so they are not angles of right triangles. (That is the coordinate plane definition of the trig functions.)
Important:
Now that we are temporarily leaving the world of right triangles, we will be thinking of an angle as a rotation. The rotation rotates one side of the angle, called the initial side, over to the other side of the angle, called the terminal side.
The angle is positive if the rotation is counter-clockwise, and is negative if the rotation is clockwise. I will indicate this direction of rotation by putting a curved arrow near the vertex.
An angle (rotation) in the coordinate plane is in standard form if its vertex is at the origin $(0,0)$, and its initial side is along the positive x-axis (horizontal axis).
For the sake of what we will be doing in the rest of this course, it is useful (and will be necessary) to refer to the horizontal axis (rather than x-axis) and the vertical axis (rather than y-axis). One big reason for this is that we will often use the variable $x$ to refer to the angle (in radians) – see further on.
Radian measure of angles (rotations): For simplicity I will define radian measure in the setting of angles in standard position in the coordinate plane.
Starting with some angle $\theta$ which is in standard position, draw (or imagine) a circle of radius $r$ centered at the origin. As we travel around this circle from the initial side to the terminal side of our angle, that marks out an arc on the circle, which is called the subtended arc for the angle $\theta$. Then the radian measure of $\theta$ is defined as the ratio $\frac{\text{length of the subtended arc}}{r}$.
For example, consider the case $\theta$ = a right angle. This angle encloses an arc whose length is $\frac{1}{4}$ of the circumference, which is $\frac{1}{4}(2\pi r) = \frac{\pi r}{2}$. So the radian measure of the right angle is this length divided by $r$:
$\frac{\left(\frac{\pi r}{2}\right)}{r} = \left(\frac{\pi r}{2}\right)\left(\frac{1}{ r}\right) = \frac{\pi}{2}$
So a right angle has radian measure $\frac{\pi}{2}$.
If you think about this example, it shows that the radian measure does not depend on the choice of the radius $r$, so we might as well use a circle of radius 1. This is called the unit circle.
The circumference of the unit circle is $2\pi$. From this we can easily see that a rotation one time around in the counter-clockwise direction has radian measure $2\pi$ (because $r=1$).
A “straight angle” is a rotation halfway around the circle, so it has radian measure $\pi$.
We already know that a right angle has radian measure $\frac{\pi}{2}$, and that makes sense also because it is half of a straight angle.
Half a right angle, which is the acute angle that appears in the isosceles right triangle, has radian measure $\frac{1}{2}\cdot \frac{\pi}{2} = \frac{\pi}{4}$.
Here are two more important angles:
The angle which is in the equilateral triangle is $\frac{1}{3}$ of a straight angle (because all 3 angles must add up to a straight angle, and they are all equal in an equilateral triangle). So the angle of an equilateral triangle is $\frac{\pi}{3}$.
This is the larger of the two acute angles in our “half-equilateral” right triangle. The smaller one is half of that, so is $\frac{1}{2}\cdot \frac{\pi}{3} = \frac{\pi}{6}$.
Become very familiar with those angles and their radian measure, preferably without trying to translate or even think of degrees! You are learning a second language, and it will harm your progress if you keep translating back into your first language. You should learn to think in radians, which are a much more important way to measure in terms of the appli9cations of trig functions. You want to immediately associate a visual image of an angle with its radian measure. Discipline!
(To be continued…)
