An angle - why an angle is the ratio of two sides of a triangle?

[Bashyboy]

An angle -- why an angle is the ratio of two sides of a triangle?

Perhaps I should have already learned this, but I just can't seem to grasp why an angle is the ratio of two sides of a triangle, or the length of a circular arc by its radius.Why is this so?

 

[mathman]

Perhaps I should have already learned this, but I just can't seem to grasp why an angle is the ratio of two sides of a triangle, or the length of a circular arc by its radius.Why is this so?

The ratio of two sides of a right triangle is a trig. function of the angle, not the angle itself.

In a circle, the angle is directly proportional to the arc length. To make a lot of mathematics simpler, the radius is used as the proportionality constant to define the units for angle, so a right angle (90 deg.) is π/2.

 

[tiny-tim]

Hi Bashyboy!

… an angle is the ratio of … the length of a circular arc by its radius.Why is this so?

The length of a circular arc of angle 2π (ie the whole circumference) is defined as 2πr.

Since the length of a circular arc of angle θ must be θ/2π times that, it is (θ/2π)2πr, = θr.

… the ratio of two sides of a triangle

sorry, no idea what you mean

 

[Vorde]

I think I understand what the OP is asking: he's wondering why sin(\theta)=one side of a triangle/the other.

The answer is (assuming I am correct in the question) is that the angle is not equal to the ratio of the two sides of a triangle. What is true is that there is a constant function that when inputted with an angle, gives the ratio of the other two sides of a right triangle with that angle. That function is sine (or cosine, or tangent: depending on which sides of the triangle you are talking about).

Instead of asking why it is those functions, realize that those functions are defined to be those specific functions that satisfy that original condition.

 

[HallsofIvy]

Perhaps I should have already learned this, but I just can't seem to grasp why an angle is the ratio of two sides of a triangle, or the length of a circular arc by its radius.Why is this so?

No, you shouldn't have "already" learned this- you shouldn't have learned it at all- it's not true. An angle is NOT "the ratio of two side of a triangle" though there are certain functions of the angle that are. Since in Euclidean geometry the measures of the angles in a triangle must add to 180 degrees, if we are talking about right triangles then given another angle, \theta, the third angle is fixed, 180- 90- \theta= 90- \theta, so, by "similar triangles" ratios of the sides are fixed and are useful enough to be given names like "sine", "cosine", etc.

It is not true that an angle is "the length of a circular arc by its radius" but it is true that the radian measure of an angle is defined as "the length of a circular arc divided by its radius.

 

[algebrat]

Perhaps I should have already learned this, but I just can't seem to grasp why an angle is the ratio of two sides of a triangle, or the length of a circular arc by its radius.Why is this so?

After you've learned all the other stuff that people have mentioned you will learn that for small angles, the angle is close to the ratio of the opposite side to one of the sides adjacent to the angle. This is important for many applications.But this fact is introduced about a year after you've learned about trigonometry, which gives the relation between angles and sides of triangles (pythagorean and standard trig functions for pairs of sides of a right triangle, law of sines and and cosines for other triangles?). And s=rθ works for sectors, which are only close to bieng triangles for small θ. Notice θ must be measured in radians. And think about the case r=1 to help develop your understanding of radians.