How do you express $cos (\frac{4 π}{3}) \cdot cos (\frac{7 π}{4})$ $cos( (4 pi)/3 ) * cos (( 7 pi) /4 )$ without using products of trigonometric functions?

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How do you express $cos (\frac{4 π}{3}) \cdot cos (\frac{7 π}{4})$ $cos( (4 pi)/3 ) * cos (( 7 pi) /4 )$ without using products of trigonometric functions?

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Answer 1 · 2016-10-04T09:33:00

$cos (\frac{4 π}{3}) cos (\frac{7 π}{4}) = \frac{1}{2} cos (\frac{13 π}{12}) + \frac{1}{2} cos (\frac{5 π}{12})$ $cos((4pi)/3)cos((7pi)/4)=1/2cos((13pi)/12)+1/2cos((5pi)/12)$

Explanation:

As $cos (A + B) = cos A cos B - sin A sin B$ $cos(A+B)=cosAcosB-sinAsinB$ and

$cos (A - B) = cos A cos B + sin A sin B$ $cos(A-B)=cosAcosB+sinAsinB$

$cos (A + B) + cos (A - B) = 2 cos A cos B$ $cos(A+B)+cos(A-B)=2cosAcosB$

or $cos A cos B = \frac{1}{2} [cos (A + B) + cos (A - B)]$ $cosAcosB=1/2[cos(A+B)+cos(A-B)]$

Hence $cos (\frac{4 π}{3}) cos (\frac{7 π}{4})$ $cos((4pi)/3)cos((7pi)/4)$

= $\frac{1}{2} [cos ((\frac{4 π}{3}) + (\frac{7 π}{4})) + cos ((\frac{4 π}{3}) - (\frac{7 π}{4}))]$ $1/2[cos(((4pi)/3)+((7pi)/4))+cos(((4pi)/3)-((7pi)/4))]$

= $\frac{1}{2} [cos (\frac{4 \times 4 π + 3 \times 7 π}{12}) + cos (\frac{4 \times 4 π - 3 \times 7 π}{12})]$ $1/2[cos((4xx4pi+3xx7pi)/12)+cos((4xx4pi-3xx7pi)/12)]$

= $\frac{1}{2} [cos (\frac{16 π + 21 π}{12}) + cos (\frac{16 π - 21 π}{12})]$ $1/2[cos((16pi+21pi)/12)+cos((16pi-21pi)/12)]$

= $\frac{1}{2} [cos (\frac{37 π}{12}) + cos (\frac{- 5 π}{12})]$ $1/2[cos((37pi)/12)+cos((-5pi)/12)]$

= $\frac{1}{2} [cos (2 π + \frac{13 π}{12}) + cos (\frac{5 π}{12})]$ $1/2[cos(2pi+(13pi)/12)+cos((5pi)/12)]$

= $\frac{1}{2} cos (\frac{13 π}{12}) + \frac{1}{2} cos (\frac{5 π}{12})$ $1/2cos((13pi)/12)+1/2cos((5pi)/12)$

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How do you express $cos (\frac{4 π}{3}) \cdot cos (\frac{7 π}{4})$ $cos( (4 pi)/3 ) * cos (( 7 pi) /4 )$ without using products of trigonometric functions?

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Explanation:

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How do you express cos(4π3)⋅cos(7π4)cos( (4 pi)/3 ) * cos (( 7 pi) /4 ) without using products of trigonometric functions?

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How do you express $cos (\frac{4 π}{3}) \cdot cos (\frac{7 π}{4})$ $cos( (4 pi)/3 ) * cos (( 7 pi) /4 )$ without using products of trigonometric functions?