Why do the ratios of sine and cosine depend upon the angle being a part of a right triangle?
because that is the way they are defined.
You cannot have a hypotenuse without a right triangle.
Read about the unit circle and the function definitions.
The ratios of sine and cosine depend upon the angle being a part of a right triangle due to the definition of these trigonometric functions.
To understand this, let's consider a right triangle with one of its acute angles being θ:
/|
/ |
/ |
Hypotenuse | Opposite Side
/ |
/ |
/θ |
/______|
Adjacent Side
In this triangle, we have three sides: the hypotenuse, the opposite side, and the adjacent side. The ratios of sine and cosine are defined based on the lengths of these sides.
The sine of an angle θ is defined as the ratio of the length of the opposite side to the length of the hypotenuse:
sin(θ) = opposite side / hypotenuse
The cosine of an angle θ is defined as the ratio of the length of the adjacent side to the length of the hypotenuse:
cos(θ) = adjacent side / hypotenuse
Now, why do these ratios depend on the angle being a part of a right triangle? The reason is that these definitions are derived from the concept of similar right triangles.
If we consider two right triangles that are similar (i.e., they have the same shape but possibly different sizes), the ratios of the corresponding sides will be the same. This property allows us to define sine and cosine for any angle, not just those in right triangles.
However, in a right triangle, the ratios of sine and cosine have a simple geometric interpretation because all three sides are directly related to the angle θ. In other words, the ratios of sine and cosine depend on the angle being a part of a right triangle because these ratios are derived from the particular lengths of sides in a right triangle.