I don’t have all that much time; I spend most of it studying. I haven’t looked up the dangers of certain plastic used for water bottles, but I have heard that #1 is one of the unsafe plastics, one that’s used commonly. Normally, to check the type of plastic a bottle is made of, you only have to flip the bottle over and look underneath at the number written inside a triangle. I read today that plastic #1 should not be used again and again; if you have a water bottle made of plastic #1, you should not continually refill the bottle.
Well that sucks, because I was reusing my plastic water bottles and they were just about my preferred water container. Through school research projects, I’ve been learning about the horrors companies would engage in, companies that sprang up relatively recently. During the Spanish-American war, the company Armour & Co. was deliberately selling US soldiers rotten meat in cans. Bolic acid, also harmful for human consumption, was also put in those cans—acting both to preserve the meat and mask the rotten scent. This company was not the only company that engaged in inhumane practices for the sake of profit.
Long story short, I don’t trust companies to want the best for me anymore. Anyway, today I went shopping with my mom, and we bought a pack of water bottles. I looked underneath the bottles, but didn’t see any triangle anywhere. The only triangle I saw was, nice and clear, on the bottom of the plastic wrapping the bottles.
There was a 4 inside the triangle. Huh, I thought, that’s great! After all, plastic #4 isn’t harmful for us (to my knowledge). Well, what do you know, I went home and took out one of the bottles. Wanting to be sure (better safe than sorry), I searched the bottle for a number in a triangle.
And I found one. On the side of the bottle, near the bottom, was a #1 in a triangle. How deceiving. Here are pictures:
Anyway. Let me show you how to answer a problem. Suppose you have this, taken from the book I myself am learning from (‘Precalculus with Limits’ by Ron Larson):
You have to find y, when y = arcsin x. In this case, x is -1/2, and ‘arcsin’ just means ‘sine inverse.’ To find out the answer, we need to find the angle which has a sin value of -1/2.
Before making any attempt to locate this angle, know that the angle must have a value between -π/2 and π/2. π/2 is 90° written in radians.
(Unfortunately, my teacher isn’t the best, choosing speed over mastery. (Sal Khan, a teacher himself in a sense, thinks the education system should change, and that mastery should be most important.) As a result I don’t understand this perfectly well, though I plan on doing so. I’ve just given up on building knowledge structures from the base up; I’ve given up on learning everything in complete order. My summarizing and explaining is a little selfish, because it allows me to go over what I’ve learned, solidify the information, and have it stored for future access.)
The unit circle looks like this. It is divided into 4 quadrants. The top right “pizza slice” is Q1 (quadrant 1); the top left is Q2; the bottom left is Q3; the bottom right is Q4. The unit circle has its center at the origin: (0,0).
Whenever there is a point on the unit circle, that point’s x-coordinate is the cosine value of the central angle corresponding to that point. A central angle is an angle on the circle. There are 4 “lines” sticking out of the circle: they are the x- and y- axes. The x-axis is horizontal and the y-axis is vertical. These two axes cut the circle into 4 equivalent slices, called quadrants.
Angles in standard position have two sides. There are two lines, and the angle exists from one line to another. The initial side is the name given to the side that rests over the x-axis. The terminal side rests wherever. It doesn’t have to rest on an axis: it can rest between two axes. That terminal side is where the angle ends.
The sin and cos values of certain angles are given by the coordinates of the point which the terminal side intersects. The x-coordinate is the cos value; the y-coordinate is the sin value. And how to find the tangent? Divide the sin value by the cos value.
tan θ = sin θ / cos θ
Let me show you.
My teacher didn’t show me how to figure out the coordinates matching certain angles: he told me to memorize the coordinates. You can use a unit circle like this one for guidance:
The image shows certain degree measures and what they are when expressed in radians. For example, 60° is shown to be π/3 radians.
Back to the problem, now that you have enough background knowledge to understand my reasoning. If we are looking for an angle that has a sin value of -1/2, and that angle cannot fall past 90° or -90° in the unit circle, then Q2 and Q3 are off-limits. Right? If the angle were in either Q2 or Q3, it would be past the 90°/ -90°. (Locate the 90° on the unit circle. That is positive because of the direction in which it has fallen. The 90° has its initial side on the x-axis and its terminal side on the y-axis. Its terminal side “opened” in the counter-clockwise direction. That makes it positive. Angles that “fall” in the clockwise direction are negative (not shown on the unit circle). To find -90° you must understand this. The initial side of 90° lies on the x-axis; the terminal side is on the y-axis, but the portion of the y-axis on the bottom half of the circle. The terminal side “opened” in the clockwise direction, coming to a stop at the portion of the y-axis that is underneath the x-axis.)
The angle, which is still unknown to us, can be either in Q1 or Q4, agree? Now it’s time for CST. CST stands for Cos Sin Tan. In Q1, the CST is +++; that is to say, in Q1, the cos, sin, and tan values are positive for all angles in that quadrant.
In Q2, CST is -+-, which means that in Q2, the cos and tan values are negative, while the sin values for all the angles in that quadrant are positive.
The CST for Q3 is –+, and the CST for Q4 is +–.
How does this apply? Let me show you. We’re looking for an angle, right? An angle that has a sin value of -1/2. This angle is either in Q1 or Q4, but judging from the CST of Q1 and Q4, Q1 can only have positive sin values. (Its CST is +++.) So it can’t be in Q1, you see: it must be in Q4.
The angle we are looking for is in Q4, though its value is yet unknown. It has a sin value of -1/2. Sine values are revealed by the y-coordinate of angles’ coordinate points. So look to the unit circle to find the angle.
The only angle in Q4, using the unit circle, which has a y-coordinate of -1/2, is 330°. Now here’s the part I don’t understand: why the author converted the answer to radians, and why they made the angle negative. Changing the direction of the angle changes its magnitude, though it is in the same area: now it becomes -30°, since we move 30° from the initial side to the terminal side in a clockwise direction.
To convert to radians (or degrees), remember this. Remember the simple fact that can be found even by using the unit circle:
π radians = 180°.
(This is not the only fact you can use, by manipulation, to convert to degrees or radians. This is simply my go-to fact.)
We have -30° and we want to convert it to radians. So we can manipulate our fact.
π radians/180° = 1°
Now what happens if we multiply both sides by -30? We get “-30°” on one side, and the radian equivalent on the other.
You get -π/6 radians if you do it right, and that is the author’s answer to the math problem.