Sine/cosine sum and difference formulas

In summary, all of the proofs I've found for formulas for any angle use angles between 0 and 2pi. For angles less than 0 or greater than 2pi there is a corresponding angle between 0 and 2pi. To find the proof for this equation, I used euler's equation, triangles, and the trigonometric circle. My solution was that cos(θ+ψ) = cos θ cos ψ - sin θ sin ψ, where θ, ψ are the angles between θ and ψ. This later simplifies to cos({θ}-{ψ}) = cos θ Cos ψ - sin θ Sin ψ. This means that sin(θ
  • #1
Werg22
1,431
1
How to proove those formulas for any angle? So far all the proofs I've found are for angles between 0 and 2pi...
 
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  • #2
Surely you know that for any angle less than 0 or greater than 2pi there is a corresponsing angle between 0 and 2pi...

If the angle x is less than 0, Use 2pi-x.
If the angle x is greater than 2pi, Use 0+x
 
  • #3
Werg22 said:
How to proove those formulas for any angle? So far all the proofs I've found are for angles between 0 and 2pi...
...which ones did you find?
 
  • #4
Sorry, i meant pi/2... (thus maximum of pi for the resultant angle)

I found one with euler's equation, another with triangles and the other with the trigonometric circle.

My solution was;

[tex]\sqrt{(\cos({{\theta} \pm {\sigma}}) - \cos{\theta})^{2} + (\sin({{\theta} \pm {\sigma}}) - \sin{\theta})^{2}} = \sqrt {({cos{\sigma} - 1})^{2} - \sin^{2}{\sigma}} [/tex]

And this later simplifies to

[tex]cos({{\theta} \pm {\sigma}}) = \cos{\theta}\cos{\sigma} \pm \sin{\theta}\sin{\sigma}[/tex]

But now I can't go further...
 
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  • #5
Alright I think I thought of a good proof

With this identity (this is using the distance formula and the coordinates on the trigonmetric circle)

[tex]\sqrt{(\cos({{\theta} \pm {\sigma}}) - \cos{\theta})^{2} + (\sin({{\theta} \pm {\sigma}}) - \sin{\theta})^{2}} = \sqrt {({cos{\sigma} - 1})^{2} - \sin^{2}{\sigma}} [/tex]

We can derive

[tex]\sin(\theta \pm \sigma) = \cos \sigma \sin \theta \pm \cos \theta \sin \sigma [/tex]

With this identity

[tex]\sin -a = -\sin a [/tex]

We can compare [tex]\sin(\theta - \sigma) [/tex] and [tex]\sin(\sigma - \theta) [/tex] As being opposite.

[tex](\sin(\theta \pm \sigma) = \cos \sigma \sin \theta \pm \cos \theta \sin \sigma) =-(\sin(\sigma \pm \theta) = \cos \theta \sin \sigma \pm \cos \sigma \sin \theta) [/tex]

The only possible solution is

[tex]\sin(\theta - \sigma) = \cos \sigma \sin \theta - \cos \theta \sin \sigma [/tex]

and

[tex]\sin(\sigma - \theta) = \cos \theta \sin \sigma - \cos \sigma \sin \theta[/tex]And since sin (a+b) is not equal to sin(a-b), exept if one of the angle is pi, thus

[tex]\sin(\theta + \sigma) = \cos \sigma \sin \theta + \cos \theta \sin \sigma [/tex]

The result stays the same if one of the angle is pi/2.

The expansion for cos(a+b) and cos (a-b) is easy to derive once we have established the expansion for sin.

Q.E.D.?
 
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  • #6
Draw the sin graph, notice the symmetry, all's well.
 

1. What are the sine/cosine sum and difference formulas?

The sine/cosine sum and difference formulas are mathematical equations that allow us to find the sine and cosine values of the sum or difference of two angles without having to calculate them separately. These formulas are used in trigonometry to simplify complex calculations and solve problems involving trigonometric functions.

2. How are the sine/cosine sum and difference formulas derived?

The sine/cosine sum and difference formulas can be derived from the sum and difference identities of trigonometric functions. These identities state that the sine and cosine of the sum or difference of two angles can be expressed in terms of the sine and cosine of the individual angles. By rearranging these identities, we can obtain the sine/cosine sum and difference formulas.

3. What are the applications of the sine/cosine sum and difference formulas?

The sine/cosine sum and difference formulas are used in various fields such as engineering, physics, and navigation. They are particularly useful in solving problems involving periodic functions, such as sound waves, light waves, and electromagnetic waves. These formulas are also used in calculating the distances and angles of objects in space.

4. How do I use the sine/cosine sum and difference formulas in calculations?

To use the sine/cosine sum and difference formulas, you need to know the values of the individual angles. You can then substitute these values into the formulas and simplify to find the sine and cosine of the sum or difference of the angles. It is important to use the correct signs for the angles, depending on whether they are in the first, second, third, or fourth quadrant.

5. Are there any special cases or exceptions for using the sine/cosine sum and difference formulas?

Yes, there are a few special cases to keep in mind when using the sine/cosine sum and difference formulas. If the angles are complementary (add up to 90 degrees), the formulas simplify to the Pythagorean identities. If the angles are supplementary (add up to 180 degrees), the formulas become negative of each other. Additionally, if the angles are equal, the formulas reduce to the double angle identities.

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