How do you find the exact value of $cos 40 sin 55 - cos 55 sin 40$ $cos40sin55-cos55sin40$ using the sum and difference, double angle or half angle formulas?

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How do you find the exact value of $cos 40 sin 55 - cos 55 sin 40$ $cos40sin55-cos55sin40$ using the sum and difference, double angle or half angle formulas?

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Answer 1 · 2016-10-31T13:05:04

$\frac{\sqrt{3} - 1}{2 \sqrt{2}}$ $(sqrt3-1)/(2sqrt2)$

Explanation:

Use trigonometric identity:
$cos (a + b) = cos a cos b - sin a sin b$ $cos(a+b)=cosacosb-sinasinb$ , we have
$sin (55 - 40) = sin 55 cos 40 - sin 40 cos 55 = sin 15$ $sin(55-40)=sin55cos40-sin40cos55 =sin15$

Find $sin 15$ $sin15$ , by using trigonometric identity:

$2 {sin}^{2} (15) = 1 - cos 30 = 1 - \frac{\sqrt{3}}{2} = \frac{2 - \sqrt{3}}{2}$ $2sin^2(15)=1-cos30=1-sqrt3/2=(2-sqrt3)/2$
${sin}^{2} (15) = \frac{2 - \sqrt{3}}{4}$ $sin^2(15)=(2-sqrt3)/4$
$sin (15) = \pm \frac{\sqrt{2 - \sqrt{3}}}{2}$ $sin(15)=+-sqrt(2-sqrt3)/2$

Only the positive value is accepted because sin15 is positive.

Finally,
$cos 40 . sin 55 - cos 55 . sin 40 = sin 15 = \frac{\sqrt{2 - \sqrt{3}}}{2}$ $cos 40.sin 55 - cos 55.sin 40 = sin 15 = sqrt(2-sqrt3)/2$

= $\frac{\sqrt{4 - 2 \sqrt{3}}}{2 \sqrt{2}}$ $(sqrt(4-2sqrt3))/(2sqrt2)$

= $\frac{\sqrt{{(\sqrt{3})}^{2} + 1^{2} - 2 \sqrt{3}}}{2 \sqrt{2}}$ $(sqrt((sqrt3)^2+1^2-2sqrt3))/(2sqrt2)$

= $\frac{\sqrt{3} - 1}{2 \sqrt{2}}$ $(sqrt3-1)/(2sqrt2)$

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How do you find the exact value of $cos 40 sin 55 - cos 55 sin 40$ $cos40sin55-cos55sin40$ using the sum and difference, double angle or half angle formulas?

1 Answer

Explanation:

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How do you find the exact value of $cos 40 sin 55 - cos 55 sin 40$ $cos40sin55-cos55sin40$ using the sum and difference, double angle or half angle formulas?