So What are Sine, Cosine, and Tangent Really?
The Fact that You are Here is the Sine of a Potential Wizard!
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What Even Are These?
So, you’ve probably heard that sine, cosine, and tangent are important. Have you ever wondered why and what they even are? In this little article, I hope to explain to you what these functions are and what they mean.
First, I would highly recommend checking out my article on radians, because if you don’t know about radians, the rest of this will be very confusing. You can find the article right here.
Sine, Cosine, and Tangent are function that take in an angle θ and output a value. You can use these for many different problems. One such problem is this:
You are standing 120 feet away from a tree and looking at the top of the tree at a 15 degree angle. How tall is the tree?
This may sound impossible, but it’s actually quite elementary. The answer is about 32.1538532147 feet. I’ll talk about how I got that at the end of the article. Just know it uses one of these functions.
Let’s start out with some general notation. The way you write the function in practice where θ is your angle in radians is:
for the sine of θ,
for the cosine of θ, and
for the tangent of θ.
How do we even Calculate Sine? I’ll give you a hint. It’s not with wizards.
It’s with Necromancers.
Formula for Sine, Cosine, and Tangent using Triangles
Now what exactly are sin, cos, and tan? To answer this, let’s imagine a right triangle with sides 3, 4, and 5. This is a famous Pythagorean triplet, and we can calculate the sine, cosine, and tangent of an angle θ. Let’s assign θ to be the angle 4:5 on this triangle. This is approximately 36.87°. Using our formula to convert to radians, we get approximately 0.6435029 radians as our angle θ.
Okay, we have our angle θ, but how will we find the sine, cosine, and tangent of these? First, let’s label our triangle lengths a, o, and h.
a is the side adjacent to our angle θ, o is the side opposite to our angle θ, and h is the hypotenuse of our triangle. Now that this is clear, we can get the formula for sine, cosine, and tangent.
This can be remembered with a mnemonic:
SohCahToa
For the sine of θ, we divide the opposite side by the hypotenuse, for the cosine of θ, we divide the adjacent side by the hypotenuse, and for the tangent of θ, we divide the opposite side by the adjacent one. Let’s just add the length 3, 4, and 5 to our formulas.
From this, one can conclude that the sine of θ is about 0.6, the cosine of θ is about 0.8, and the tangent of θ is about 0.75. These are the correct values of a 3 4 5 triangle. If you put this value of θ into a calculator, the results would be very slightly off because the angle θ is only an approximation for the angle we were computing.
Sine, Cosine, and Tangent Formula using Unit Circle
let’s imagine a unit circle with radius one. Let’s have an angle θ and work our way from there.
Okay, it looks a little crazy, but I promise it will make sense in the end. We have our angle θ. At the point where it hits the arc of the circle, the sine value of θ is just the y-value of that point, and the cosine of θ is just the x-value. The tangent of θ is a little more complicated. The way it works is we draw a line perpendicular to the line created by θ, as you can see in the picture. We then extend this line until the y-value of the line = 0, the y-value of the base. The length of that new line is equal to the tangent of θ. We can see that sin(θ) is limited from -1 to 1, and the same is true for cos(θ). The Tangent, however, can get much bigger than that. It can actually approach infinity, as we will see in the next section.
Fun Fact About Tangent
So you now know 2 different ways to get sine, cosine, and tangent. However, tangent is extremely funky. This is because the tangent of θ is the sine of θ divided by the cosine of θ. What happens when we set θ to be 90°, or π/2 radians?
We can see from our circle diagram that sin(π/2) = 1, and that cos(π/2) = 0. Therefore the tangent must be undefined!
We can check this by plotting it on our unit circle.
As you can see, the tangent line goes on forever, never stopping. This is really interesting, isn’t it?
Using the same form of reasoning, try to find tan(3/4π)
The Problem from the Beginning
Obviously, this was just a quick overview of sine, cosine, and tangent. Let’s go back to the problem from the beginning.
You are standing 120 feet away from a tree and looking at the top of the tree at a 15 degree angle. How tall is the tree?
The way you solve this is using the tangent. Let’s label the height of the tree s, and try to solve this. Our angle θ is 15 degrees, our about 0.261799 radians. We know that tan(θ) is about s/120, because our adjacent angle is 120 feet and our opposite angle is s.
We can find the solution for tan(θ) as well.
We know that s/120 = tan(θ), and tan(θ) is about 0.26794919243, so we can multiply tan(θ) by 120 to get s!
There you go, that is the value of s, or the height of the tree!
There you go, that’s a brief overview on sine, cosine, and tangent.
-The Math Necromancer
:)
The Fact that You are Here is the Sine of a Potential Wizard!