Solving for Cosine & Sine Sum/Difference of Two Angles: Confusing?

[Miike012]

I have yet to see or understand the benefit of knowing the cosine and sine sum and difference of two angles identity... can someone please explain why you would want to break up one angle into two angles... then solve from there... does not make sense to me... just use a calculator.

 

[blather]

Deriving the rotation matricies.

 

[jhae2.718]

You'll find that many of the trig identities are mathematically useful in ways beyond simple calculation.

 

[Miike012]

Because as I was looking at the the identity and the proof.. I noticed that only one of the two angles were one of the "special angles" given on the unit circle... you can solve for the two angles without using the idenetity... so for me it just seems useless...
for instance... let's say we have two angles 30 and 45... you could either use the identity or just straight out solve for it...
not let's say we have two angles 1 and 45... you can not solve for this using the sum of two angles identity... HOWEVER you can solve this using algebra... and law of sines...

 

[Miike012]

Sorry... I ment to say you can't solve without a calc unless u know what sin or cos of angle 1 deg. is... and sorry... as I was looking at it,... you can not solve algebraically by useing law of sines.. if u don't know what sin or cos of angle one is either... so both methods in my mind seem equal.. however the identity might be a tad quicker... but if your looking for quickness just use a calc...

 

[jhae2.718]

For the sum/difference, formulas, as blather stated, they can be used to derive the rotational matrix in Euclidean space: http://planetmath.org/encyclopedia/DerivationOfRotationMatrixUsingPolarCoordinates.html

 

[Miike012]

Alright.. well I have no idea what that is.. I am only in trig... when will I get there?

 

[jhae2.718]

Linear algebra, it's a little ways off for you then.

When you take calc, for another example, you'll find that $$\cos^2(\theta) = \frac{1+\cos(2\theta)}{2}$$ is useful for integrals. (You can't integrate cos^2 or sin^2, but you can transfrom them inot something linear and then solve it.)

Trig seems pointless when you just calculate the angles like that, but it's actually extremely useful.

 

[Miike012]

Lol yes I have some time before I take that class. lol

 

[blather]

Miike012, don't think that your classes are holding you back. If you'd like to poke around in a library, then check out a calculus book, muddle through that for a bit, and then get a book called:

Basic Complex Analysis
by Marsden

This book shows a lot of cool things about trig identities in one of the first two chapters. You are always free to go faster than your education.

 

[Miike012]

Thank you for the advise blather... I will also check out the book...