Stony Brook MAT 123 Fall 2015
Definition of Sine and Cosine (in Right Triangles)

Start   This module will be about where SINE and COSINE come from.
Now, if you remember from geometry, two triangles are similar if all the angles are equal. Now they are similar in other ways but, if all the angles are equal -congruent- then the two triangles are similar and that means that their sides are proportional which means that the ratio of this side to this side is the same as
0:30the ratio of this hypotenuse to this hypotenuse.
So how do we know this two triangles are similar? Well, they both have a right angle, and lets say this angle is the same, let's says that is 20 degrees. Well, the third angle is automatically going to be equal because they have to add up to 180 degrees. So that's 20 degrees and 90 degrees makes 110 degrees so that is 70 degrees and that's 70 degrees. And then we know that the ratio of A to B
1:03would be the same of the ratio of C over D and the same as the ratio of E over F.
So we have a right triangle and we know the angle is 20 degrees then, any other right triangle that has a 20 degree angle would have another triangle whose angle is 70 degrees. And therefore would be similar to any other triangle and these ratios will always be the same.
1:32We give a name to these ratios.
The name of the ratios are "SINE", "COSINE" and "TANGENT".
So let's draw a new triangle.
We had a triangle and let's call this angle x so there are 3 sides to this triangle: A, B and C.
And we know that the ratio of A to C in this triangle
2:02would be same as for any other right triangle that as an angle of x. So we name that ratio "SINE" (sin). We say that the sine of x is side A divided by side C. The other angle is complementary, so we use another word "COSINE" (cos). And we say that cosine of x is the other side divided by C
2:34And the "TANGENT" (tan) is the third ratio. And the "TANGENT" of x would be side A divided by side B. And we write that as tan(x) is A over B. So let's do an example.
If we call this angle x and this is 3 and
3:01this is 4 and that's 5 then, the sine of x is 3 over 5.
Any triangle, any right triangle with that x, the sine would be 3 over 5.
The cosine is 4 over 5 and the tangent is 3 over 4.
3:33And that is the definition of "SINE" and "COSINE".