Sum and Difference Identities
Key Questions
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Main Sum and Differences Trigonometric Identities
#cos (a - b) = cos a*cos b + sin a*sin b#
#cos (a + b) = cos a*cos b - sin a*sin b#
#sin (a - b) = sin a*cos b - sin b*cos a#
#sin (a + b) = sin a*cos b + sin b*cos a#
#tan (a - b) = (tan a - tan b)/(1 + tan a*tan b)#
#tan (a + b) = (tan a + tan b)/(1 -tan a*tan b)# Application of Sum and Differences Trigonometric Identities
Example 1: Find
#sin 2a# .#sin 2a#
#= sin (a + a)#
#= sin a*cos a + sin a*cos a#
#= 2*sin a*cos a# Example 2: Find
#cos 2a# .#cos 2a#
#= cos (a + a)#
#= cos a*cos a - sin a*sin a#
#= cos^2 a - sin^2 a# Example 3: Find
#cos ((13pi)/12)# .#cos ((13pi)/12)#
#= cos (pi/3 + (3pi)/4)#
#= cos (pi/3)*cos ((3pi)/4) - sin (pi/3)*sin ((3pi)/4)#
#= -(sqrt2)/4 - (sqrt6)/4#
#= -[sqrt2 + sqrt6]/4# Nghi N. · 3 · Apr 7 2015
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Here is an example of using a sum identity:
Find
#sin15^@# .If we can find (think of) two angles
#A# and#B# whose sum or whose difference is 15, and whose sine and cosine we know.#sin(A-B)=sinAcosB-cosAsinB# We might notice that
#75-60=15#
so#sin15^@=sin(75^@-60^@)=sin75^@cos60^@-cos75^@sin60^@# BUT we don't know sine and cosine of
#75^@# . So this won't get us the answer. (I included it because when solving problems we DO sometimes think of approaches that won't work. And that's OK.)#45-30=15# and I do know the trig functions for#45^@# and#30^@# #sin15^@=sin(45^@-30^@)=sin45^@cos30^@-cos45^@sin30^@# #=(sqrt2/2)(sqrt3/2)-(sqrt2/2)(1/2)# #=(sqrt6 - sqrt 2)/4# There are other way of writing the answer.
Note 1
We could use the same two angles and the identity for#cos(A-B)# to find#cos 15^@# Note 2
Instead of#45-30=15# we could have used#60-45=15# Note 3
Now that we have#sin 15^@# we could use#60+15=75# and#sin(A+B)# to find#sin75^@# . Although if the question had been to find#sin75^@, I'd probably use # 30^@# and # 45^@#
Jim H
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Mar 25 2015