Can someone tell my what cos, sin and tan actually measure?
In: Mathematics
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.
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