Question #191f2

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Question #191f2

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Answer 1 · 2015-09-11T12:28:12

$cos 75 = \frac{1}{2} \sqrt{\frac{2 + \sqrt{3}}{4}} - \frac{\sqrt{3}}{2} \sqrt{\frac{2 - \sqrt{3}}{4}}$ $cos 75 =1/2 sqrt((2+sqrt(3))/4) - sqrt(3)/2 sqrt((2-sqrt(3))/4)$

Explanation:

You need to find a way to express the angle $75$ $75$ degrees in terms of the standard angles.

The standard angles are (in degrees):

$30$ $30$ degrees
$45$ $45$ degrees
$60$ $60$ degrees
$90$ $90$ degrees

You need to come up with an expression that equals $75$ $75$ , using only addition and dividing by 2.

One possible combination is:

$60 + \frac{30}{2} = 75$ $60 + 30/2 = 75$

Therefore, we can solve this by hand by using:

$cos 75$ $cos 75$
$= cos (60 + \frac{30}{2})$ $= cos(60+30/2)$

First we can use the sum and difference identities:

$cos (α + β) = cos α cos β - sin α sin β$ $color(green)(cos (alpha + beta) = cos alpha cos beta - sin alpha sin beta)$

$cos (60 + \frac{30}{2})$ $cos(60+30/2)$

$= cos 60 cos (\frac{30}{2}) - sin 60 sin (\frac{30}{2})$ $= cos 60 cos (30/2) - sin 60 sin (30/2)$

Then we can use the half angle identities:

$sin (\frac{θ}{2}) = \sqrt{\frac{1 - cos θ}{2}}$ $color(green)(sin (theta/2) = sqrt((1-cos theta)/2))$

$cos (\frac{θ}{2}) = \sqrt{\frac{1 + cos θ}{2}}$ $color(green)(cos(theta/2) = sqrt((1+cos theta)/2))$

$cos 60 cos (\frac{30}{2}) - sin 60 sin (\frac{30}{2})$ $cos 60 cos (30/2) - sin 60 sin (30/2)$
$= cos 60 \sqrt{\frac{1 + cos 30}{2}} - sin 60 \sqrt{\frac{1 - cos 30}{2}}$ $= cos 60 sqrt((1+cos30)/2) - sin 60 sqrt((1-cos30)/2)$

Now we can evaluate the expression by hand:

$cos 60 = \frac{1}{2}$ $color(green)(cos 60 = 1/2)$

$sin 60 = \frac{\sqrt{3}}{2}$ $color(green)(sin 60 = sqrt(3)/2)$

$cos 30 = \frac{\sqrt{3}}{2}$ $color(green)(cos 30 = sqrt(3)/2)$

$cos 60 \sqrt{\frac{1 + cos 30}{2}} - sin 60 \sqrt{\frac{1 - cos 30}{2}}$ $cos 60 sqrt((1+cos30)/2) - sin 60 sqrt((1-cos30)/2)$

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

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

$= (\frac{1}{2}) \sqrt{\frac{2 + \sqrt{3}}{4}} - (\frac{\sqrt{3}}{2}) \sqrt{\frac{2 - \sqrt{3}}{4}}$ $=(1/2) sqrt((2+sqrt(3))/4) - (sqrt(3)/2) sqrt((2-sqrt(3))/4)$

$= \frac{1}{2} \sqrt{\frac{2 + \sqrt{3}}{4}} - \frac{\sqrt{3}}{2} \sqrt{\frac{2 - \sqrt{3}}{4}}$ $color(blue)(=1/2 sqrt((2+sqrt(3))/4) - sqrt(3)/2 sqrt((2-sqrt(3))/4))$

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Question #191f2

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