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November 27th, 2016, 08:39 AM   #1
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adding subtracting multiples of 360

Hi folks

I'm doing some study into adding subtracting multiples of 360.

Something's confusing me - probably really easy but I can't figure it.

Ok in one question it asks:

Simplify the sin (x + 400) = sin (x+40)

Yet

sin (θ + 1000) = sin (θ - 80)

Why is the second one not equal to Sin (θ + 280)?

Last edited by skipjack; November 27th, 2016 at 10:44 AM.
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November 27th, 2016, 08:50 AM   #2
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$\sin(\theta+1000^\circ) = \sin{\theta}\cos(1000^\circ) + \cos{\theta}\sin(1000^\circ) = \sin{\theta}\cos(-80^\circ) + \cos{\theta}\sin(-80^\circ) = \sin{\theta}\cos(80^\circ) - \cos{\theta}\sin(80^\circ) = \sin(\theta - 80^\circ)$
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Last edited by skeeter; November 27th, 2016 at 09:05 AM.
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November 27th, 2016, 08:54 AM   #3
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Ok I'm a n00b that's even more confusing lol. I've noticed the answers are all <180. Most are <90 degrees.
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November 27th, 2016, 09:12 AM   #4
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you need to learn some identities, namely ...

sum/difference identities for sine & cosine

$\sin(a \pm b) = \sin{a}\cos{b} \pm \cos{a}\sin{b}$

$\cos(a \pm b) = \cos{a}\cos{b} \mp \sin{a}\sin{b}$

odd/even identities for sine & cosine ...

$\sin(-\theta) = -\sin{\theta}$

$\cos(-\theta) = \cos{\theta}$
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November 27th, 2016, 09:27 AM   #5
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I may have made it more difficult than necessary ... note that $1000 - 3(360) = -80$
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November 27th, 2016, 09:34 AM   #6
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Ok thanks Skeeter I'll struggle on through it. I'm liking the odds of the answer being less than 180 degrees. I shall invent a new mathematical law. No proof necessary
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November 28th, 2016, 10:13 AM   #7
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sin(n*360 + x) = sin x.
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