The Physics Philes, lesson 20: Here’s Your Sine
In which graphs are drawn, I radiate radians, and periodic functions are defined.
Welp, it looks like I’ve successfully made it through the algebra portion of my review. At least as far as learning calculus is concerned. However, I still have one more topic to review until I can get my teeth into calc: trigonometry.
I don’t have great memories of high school trigonometry. Just a lot of cursing and tears. But don’t worry. I’m marginally more mature now. I think I can handle it after all these years.
So let’s jump right in.
The past couple of weeks we’ve been reviewing functions and how they work. But there is a special type of function called a periodic function. In a periodic function, the values repeat over and over at the same rate and at the same time intervals. In other words, the graph will repeat itself after a fixed period of time. Trigonometric functions are periodic functions. The fact that periodic functions will just go on and on forever is an important one. That means that we have have an infinite number of inputs that have the same value. These are called coterminal angles. These angles have the same function value because the space between them is a multiple of the function’s period.
That’s about as clear as mud, isn’t it? Maybe a sample problem will help clear it up.
Find two angles (one positive and one negative) that have the same sine value as Π/4.
This is really pretty easy. The period value of sine is 2Π. (More on sine and the other trigonometric functions later.) So, to find one positive coterminal angle and one negative coterminal angle, we just have to add and subtract 2Π to Π/4, respectively. Let’s add first:
Π/4 + 2Π = Π/4 + 8Π/4 = 9Π/4
Remember, in order to add fractions, we need a common denominator. In this case, we needed to change 2Π to the equivalent 8Π/4. To find the negative coterminal angle, we just need to subtract 8Π/4:
Π/4 – 2Π = Π/4 – 8Π/4 = -7Π/4
See? It’s that easy. We found a positive and a negative angle that is coterminal to Π/4, and thus have the same sine value.
I mentioned earlier that trigonometric functions are periodic functions. That’s all well and good, but what are trigonometric functions? I’m glad you asked.
There are six trigonometric functions. Three are pretty familiar. The other three are a little more esoteric.
This is the graph for sine. Do you see how, in this illustration, the graph starts at the origin and starts over again at 2Π? That means it’s period is 2Π radians. The graph starts over after it has gone a period of 2Π. The range of sine is -1 ≤ y ≤ 1. That just means that it doesn’t go any higher up the y-axis than 1, and it goes no lower than -1. The domain for sine is unrestricted. The sine function has a value of 0 whenever the input is a multiple of Π. So if the input is Π, 2Π, 3Π, etc., the value of sine is 0.
The graph for cosine is actually really similar to the graph for sine; it’s just shifted Π/2 to the right:
In this graph, the blue is cosine and the red is sine. You can see how similar they are. In fact, cosine is a cofunction of sine. That means that sine and cosine have the same domain, range, and period as sine. Cosine has a value of 0 at all “half-Πs.” (For example, Π/2, 3Π/2, etc.)
Tangent is really just the quotient of sin x and cos x. Cosine is in the denominator, which means that the tangent will be undefined
when cos = 0 (which, as you’ll remember, is on the half-Πs.) Because there are, at times, zeros in the denominator, the tangent graph is riddled with asymptotes. Asymptotes are lines representing an unattainable value that shapes a graph. Vertical asymptotes, which are found in tangent graphs, usually indicate the presence of a zero in the denominator. The tangent graph crosses the x-axis at the midpoint between the asymptotes.
Because the graph is undefined at the half-Πs, the domain of tangent does not include those numbers. Its range, however, is all real numbers and its period is Π.
Now we get into the less well-known trigonometric functions. Cotangent is the cofunction of tangent; it’s the quotient of cos x and sin x. In addition, cotangent is the reciprocal of tangent, which means it can be expressed as 1 / tan x.
Just as with tangent, cotangent includes asymptotes when sine (which is in the denominator) is equal to zero, at which point the graph is undefined. This occurs at multiples of Π, which makes sense when you look at the sine graph. It crosses the x-axis at multiples of Π. The range includes all real numbers, domain includes all real numbers except multiples of Π, and the period is Π.
This trigonometric function is the reciprocal of cosine, so the graph is undefined when cos x = 0. Which means – you guessed it! – this function has vertical asymptotes. Secant is different from the graphs we’ve looked at so far because it has no x-intercepts at all.
The range of secant is y ≤ -1 and y ≥ 1, which means that the closest the graph will get to the x-axis is -1 and 1. The period of secant is 2Π.
Finally, the last of the trigonometric functions. The graph of cosecant looks bit like the graph for secant, just shifted over Π/2.
As you can see, the range of cosecant is the same as the range for secant. The closest it will get to the x-axis is -1 and 1. Cosecant is also the reciprocal of sine, so when sin x = 0, you’ll find a vertical asymptote. The domain of cosecant is all real numbers except for those half-Πs, and the period is 2Π.
That’s a lot of graphs, and they are graphs I’ll need to be familiar with if I’m going to take on calculus. But look, four of these functions are based on the sine and cosine. So as long as I know the sine and cosine in a problem, I can derive the values for the rest. In fact, let’s try it.
If cos Θ = 1/3 an sin Θ = √8/3, evaluate tan Θ and sec Θ.
If you’ll remember, tan x = sin x / cos x and sec x = 1 / cos x. All we have to do is plug in those values.
tan Θ = √8/3 / 1/3 = √8/3(3/1) = √8
sec Θ = 1 / 1/3 = 1/1(3/1) = 3
There, see? Easy. No problem at all.
There is only one more week of review before I dive into the totally unfamiliar world of calculus. That means it’s only one more week until you get to see me flail around and pull my hair out. Should be a good time.
Featured image credit: jonoakley