Sum and Difference Identities

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Math Analysis - Trigonometry - Sum and Difference Formulas - Sine - Cosine

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1 of 3 videos by AJ Speller

Key Questions

  • 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#

  • 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^@#

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Questions

  • Aviv S. answered · 2 months ago