Basic trig question: reference angles

[cupu]

Hello,

Preparing for a test, I've had to go through some very basic trigonometry and I've got to thinking why reference angles "work". I've gone through my study material and through another trigonometry book I have around the house and references angles are never proven, the theorem is just stated and then used.
Intuitively it makes sense right away; for example how sine goes from 0 to 1 as the angle goes from 0 to π/2, then as the angle increases, the reference angle in quadrant 2 has the same value as the angle in standard position. But after that I'm thinking that it only works because it's defined that way ...

Anyway, it's probably *really* easy, that's why it's not explained anywhere (that I've searched) but, starting from the basic definition of sin/cos/etc. in a right angled triangle I can't see how the theorem of reference angle is proven; any pointer is much appreciated.

Thank you very much!

 

[Simon Bridge]

But after that I'm thinking that it only works because it's defined that way ...

That is correct. Everything in mathematics boils down to the definition.
Start by stating the definition.

But the definition is that way because of the definition of what an angle is in the first place.
The definition is useful because of the periodic properties of the trig functions and the rules for addition and subtraction.
It will probably make more sense if you imagine you didnt have a calculator and you have been hired to construct a table of values for the sine of an angle for any angle ... how would you go about making your job easier?

Have you look further afield:
http://www.sparknotes.com/math/trigonometry/trigonometricfunctions/section4.rhtml
... I think the main problem with answring your question is that you have not explained what the problem with that sort of explanation is.

 

[cupu]

Hello,

Thanks for the reply. My problem is just that when I think about reference angles, they make sense because of the periodic nature of trig functions; then I think that the periodicity is there because we use reference angles to compute the values of trig functions for angles between (pi/2, 2pi) and then my brain deadlocks.
Also somewhere online - I think wikipedia, but can't find the link anymore - this was called the "Theorem of reference angles"; I expected a theorem to have a proof, but couldn't find it or deduce it.

I can easily live with the "that's the way it's defined", I just thought/think that I'm missing something very evident.

Cheers!

 

[Simon Bridge]

That's not correct.
The periodicity comes from what the trig functions are.
Reference angles are useful because of the periodicity.

eg.
The size of an angle is defined as the length of arc of the unit circle subtended by the angle.
The radien is when the unit circle is defined as that having a radius of 1.
The other way to define a unit circle is to make the circumference 1 - which most people then divide into 360 degrees.

Watch how the trig functions are defined in terms of the unit circle:
http://www.mathsisfun.com/algebra/trig-interactive-unit-circle.html
... see where the periodicity comes from? No reference angles used.

 

[CellCoree]

Hello,

Thank you for your question about reference angles in trigonometry. The concept of reference angles is indeed very important and useful in solving trigonometric problems. Let me try to explain the idea behind reference angles and why they are used in trigonometry.

First, let's review the definition of reference angle. A reference angle is the acute angle formed between the terminal side of an angle in standard position and the x-axis. In simpler terms, it is the angle between the positive x-axis and the ray that forms the angle in standard position.

Now, why do we use reference angles in trigonometry? The main reason is that it allows us to simplify calculations and make them more manageable. For example, instead of dealing with a large angle like 150 degrees, we can use the reference angle of 30 degrees and apply the appropriate trigonometric ratios to solve the problem. This makes the calculations easier and more efficient.

But why does this work? It all goes back to the definitions of the trigonometric functions - sine, cosine, and tangent. These functions are defined as the ratios of the sides of a right triangle. However, these definitions are not limited to just the first quadrant, they can be applied to any angle in the coordinate plane. When we use reference angles, we are essentially finding the corresponding angle in the first quadrant that has the same trigonometric values as the given angle. This is why the values of the trigonometric functions remain the same for the reference angle and the given angle.

To prove this concept, we can use the unit circle. The unit circle is a circle with a radius of 1 unit, centered at the origin of the coordinate plane. Each point on the unit circle represents an angle in standard position. By using the coordinates of these points, we can find the values of the trigonometric functions for any angle. When we use reference angles, we are essentially finding the coordinates of the corresponding point on the unit circle, which will have the same trigonometric values as the given angle.

I hope this explanation helps you understand the concept of reference angles better. Remember, they are a helpful tool in solving trigonometric problems, and they are based on the definitions of the trigonometric functions and the unit circle. Keep practicing and you will become more comfortable with using reference angles in your trigonometry problems. Best of luck on your test!