Can someone tell my what cos, sin and tan actually measure?
In: Mathematics
First let’s explain radiants. Just like degrees, radiants are a measure of angle. When you think of a unit circle (a circle with radius 1) the length along the edge of the circle is equal to the angle in radiants for that slice of the circle. So we know the circumference of a unit circle is 2π so 360° is 2π radiants, 180° is half a circle so π radiants, 90° covers a quarter of the circle so ½ π radiants, etc.
sin(t) is the Y value of rotating around a unit circle counterclockwise starting at (0,1) by t radiants.
cos(t) is the X value of rotating around a unit circle counterclockwise starting at (0,1) by t radiants.
tan(t) = sin(t)/cos(t)
It’s a ratio between the sides of a right triangle. Take a triangle with one 90 degree angle and choose one of the other two angles to start with. This will be our starting angle. Now, because all triangles have angles which add to 180 degrees, the sum of the other two angles must also add up to 90.
From our starting angle, we define 3 ratios: Sine, Cosine, and Tangent. Let’s name the 3 sides of the triangle. The **hypotenuse** is the longest side of the triangle, which is the side that is opposite the 90 degree angle. The remaining side closest to our starting angle is called **adjacent**. The third side, opposite our starting angle is called **opposite**.
**Sine** is defined as the ratio of the **opposite** divided by the **hypotenuse**.
**Cosine** is defined as the ratio of the **adjacent** divided by the **hypotenuse**.
**Tangent** is defined as the ratio of the **opposite** divided by the **adjacent**.
S=O/H
C=A/H
T=O/A
If you define the length of the hypotenuse to be one unit, Sine measures the height of the opposite side, while cosine measures the length of the adjacent side.
Using this **Unit Circle**, you can find the *y* and *x* coordinates on a graph of any point on that circle if you know the angle of the triangle, or if you know one of the two lengths.
Conversely, if you know the length of any two sides of the triangle, given that one angle is a 90 degree angle, you can easily find the angles using these functions, and compute the length of the third side.
This is basically all of trigonometry condensed. It takes a while to get the hang of it all.
Trigonometry is basically defining triangles based on circles, and defining circles based on triangles. If you have a little information, you can deduce the rest of the information, due to this fundamental fact that triangles always add up to 180 degrees. It’s the Pythagorean theorem with extra steps.
People have already answered this. But I will go into more detail here.
The most basic form of a geometric shape is a triangle. A triangle consists of three straight lines joined together to form three vertexes, or angles. All triangles have the following properties: 1) the sum of the three interior angles is 180⁰ (or PI radians), and 2) The sum of any two sides of the triangle must be greater than the remaining side.
For any triangle, we give the side with the greatest length, the name “hypotenuse.” If we are looking at a specific angle, then the two sides that form the angle are known as “adjacents” and the side not touching the angle is called the “opposite.”
Now, an important point is that the largest angle of a triangle is always opposite to the hypotenuse of the triangle and that the smallest angle is always opposite the smallest side. Additionally, if you have two triangles that have the same angles, then the ratio of their sides will be the same.
This brings us to the next point. A right triangle. A right triangle has one angle equal to 90⁰. Since the sum of all interior angles must be 180⁰ this means that the other two angles must be less than 90⁰. Therefore, the hypotenuse of a right triangle is the side opposite the right angle. This brings us to the Pythagorean Theorum, which states that the sum of the squares of the adjacent sides of the 90⁰ angle is equal to the square of the hypotenuse. In other words, A^2 + B^2 =C^2 where C is the length of the hypotenuse and A and B are the lengths of the other two sides.
Okay, now that the background is out of the way. Let’s actually get to the meat of your question. Remember how I mentioned that if there are two triangles with the same angles then the ratio of their sides will be the same? This is the basis of trigonometry.
So imagine a right triangle. It has an angle “a” which is adjacent to side B and side C and opposite side A. Side C is the hypotenuse. For all right triangles that contain angle “a,” the ratio of A/C will be the same, as will the ratio of B/C and A/B. To make matters simple, we give these the names Sine of angle a, cosine of angle a, and tangent of angle a, respectively. Or, more simply, SIN(a), COS(a), and TAN(a).
As an aside, remember the Pythagorean Theorum? A^2 + B^2 = C^2? Well, if we divide both sides by C^2, then we get the equation (A/C)^2 + (B/C)^2 =1. And since A/C is SIN(a) and B/C is COS(a), this is where we get the famous equation SIN(a)^2 + COS(a)^2 =1.
Sine, cosine and tangent don’t measure anything. They are math functions. You give them an angle and they give you a number back. For example, sin(90 degrees) gives you 1. The number has to do with the lengths of a right triangle. The great thing about them is, if you know the angle and the length of one of the triangle’s sides, sin, cos, and tan can tell you the length of another side. This can be very handy when building things with flat sides.
Imagine you are standing in certain place on large flat field. Lets say you are facing north. If you turn by 0 degrees and walk 1 km straight, you will end 1 km to the north.
If you instead turn 90 degrees to the right and walk 1km straight, you will end 1 km to the east and 0km to the north.
If you instead turn 45 degrees and walk straight 1km, you will be 0.7km to the north and 0.7km to the east.
What if you turn for 72 degrees and walk straight for 1 km? Where you will end? SIN and COS just answers to that.
You will be sin(72) km in the north and cos(72) km to the east.
North: sin(72) x 1km = 0.309 x 1km = 0.309 km
East: cos(72) x 1km = 0.951 x 1km = 0.951 km
So turning right 72 degrees and walking 1km straight, you will end up 309 meters to the north and 951 meters to the east from your starting point.
This brings back memories. Our Maths teacher gave us a mnemonic, that is still ingrained in my head.
“Some People Have Curly Black Hair, Turned Permanently Brown”.
Sin = Perpendicular/Hypotenuse
Cos = Base/Hypotenuse
Tan = Perpendicular/Base
I hope this helps someone.
There are three sides to a triangle, and three angles inside of the triangle that are formed by pairs of these sides.
Cos, sin, and tan are relationships between the lengths of these sides and the size of these angles. In the real world, these relationships let you calculate things (lengths or angles) that would otherwise be really hard for you to measure, by taking advantage of the fact that they are true for *all triangles*.
Don’t have a tool to measure angles? You can use cos, sin, or tan to turn a measurement of two lengths into an angle.
Don’t have a tape long enough to measure the length you care about? You can use cos, sin or tan to turn an angle measurement and a much smaller length measurement into the longer length you care about.
Once you see that much of the world’s geometry can be broken down into triangles, the real power of these relationships becomes clear. Everything from construction to astronomy benefits from these relationships.
I never appreciated trigonometry in high school. But, then I became interested in creating computer games and I found that I needed to use sin and cos in order to calculate points in 2d/3d space. I use atan to find the angle between 2 points.
Basically, if you have an angle, like the heading of an airplane, you can calculate the x and y coordinates for other points using sin and cos. It is very useful in programming with graphics.
https://en.m.wikipedia.org/wiki/File:Circle_cos_sin.gif
Stare at this and it will dawn upon you.
To a five year old: think of a fly sitting on the edge of a table fans blade. Sin and cos are machines that can tell you how far away the fly is from the middle of the fan. The machines don’t work unless you tell them the angle the fans blade makes from the surface of the table. Cos will tell you the horizontal (left or right) distance from the middle while sin tells you the vertical (up or down) distance from the middle.
Here let’s pull out a calculator because it has the sin and cos machines on it. See this table fan? Its blade is 1 ft. long. Right now the fly is sitting on the edge and the fan blade is sticking straight out to the right. So the angle this blade is making is 0 degrees. How far to the right is the fly from the middle? That’s right it’s 1ft away, here let’s use the cos machine to double check cos(0) = 1 !
Now how far above is the fly from the middle? That’s right it’s 0ft above the middle of the fan because they’re at the same height right now. Let’s check with the machine: sin(0) = 0 !
Left rotate the fan so the fly is now straight up from the middle. The fan blade is now making a 90 degree angle from the desk. How far away (left/right) is the fly to the middle how? That’s right it’s 0ft! Let’s check: cos(90) = 0. And now how far above is the fly from the middle? That’s right it’s now 1 ft. above the fly! Let’s check: sin(90) = 1
In a nut shell:
cos(angle) = horizontal distance from origin
sin(angle) =. Vertical distance from origin
cos and sin are the main functions here but we like tangent as well as it’s the ratio of sin/cos
Latest Answers