SOLUTION: Use the angle sum or difference identity to find the exact value of cos195 csc5pi/12

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Question 166485: Use the angle sum or difference identity to find the exact value of
cos195

csc5pi/12
Answer by Edwin McCravy(6938) About Me  (Show Source):
You can put this solution on YOUR website!
Use the angle sum or difference identity to find the exact value of
cos195 

You have to figure out a way to make 195° using 
the sum or difference of two special angles from this list:

30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 
225°, 240°, 270°, 300°, 315°, 330°.

You can write 195° as any one of these:

150°+45°, 135°+60°, 225°-30°, 240°-45°, 315°-120°, 330°-135°

If you choose to use the first one, 150°+45°, you then use

cos%28alpha%2Bbeta%29+=+cos%28alpha%29cos%28beta%29-sin%28alpha%29sin%28beta%29

cos%28195%29=cos%28150%2B45%29+=+cos%28150%29cos%2845%29-sin%28150%29sin%2845%29=

%28-sqrt%283%29%2F2%29%28sqrt%282%29%2F2%29-%281%2F2%29%28sqrt%282%29%2F2%29

-sqrt%286%29%2F4+-sqrt%282%29%2F4

%28-sqrt%286%29-sqrt%282%29%29%2F4

--------------------------------

csc%285pi%2F12%29

Let's convert that to degrees:

5pi%2F12=%285pi%2F12%29%28180%2Fpi%29=75°

Then we can write 75° as 30°+45°

csc%285pi%2F12%29=csc%2875%29=1%2Fsin%2875%29=1%2Fsin%2830%2B45%29=

Then we can use the identity: sin%28alpha%2Bbeta%29+=+sin%28alpha%29cos%28beta%29%2Bcos%28alpha%29sin%28beta%29

1%2Fsin%2830%2B45%29+=+1%2F%28sin%2830%29cos%2845%29%2Bcos%2830%29sin%2845%29%29=

1%2F%28%281%2F2%29%28sqrt%282%29%2F2%29%2B%28sqrt%283%29%2F2%29%28sqrt%282%29%2F2%29%29=

1%2F%28%28sqrt%282%29%2F4%29%2B%28sqrt%286%29%2F4%29%29=

1%2F%28++%28sqrt%282%29%2Bsqrt%286%29%29%2F4++%29=

matrix%281%2C3%2C+++++1%2C+%27%F7%27%2C+%28sqrt%282%29%2Bsqrt%286%29%29%2F4+%29

matrix%281%2C3%2C+++++1%2C+%27%D7%27%2C+4%2F%28sqrt%282%29%2Bsqrt%286%29%29+%29

Rationalize the denominator:

matrix%281%2C3%2C%0D%0A+4%2F%28sqrt%282%29%2Bsqrt%286%29%29++%2C%0D%0A%0D%0A+%27%D7%27%2C%0D%0A%0D%0A%28sqrt%282%29-sqrt%286%29%29%2F%28sqrt%282%29-sqrt%286%29%29++%0D%0A%0D%0A%29

Indicate multiplication of numerators and denominators:

+%284%28sqrt%282%29-sqrt%286%29%29%29+%2F%0D%0A+%28+sqrt%282%29%2Bsqrt%286%29+%29%28+sqrt%282%29-sqrt%286%29+%29+

Multiply out the bottom:

+%284%28sqrt%282%29-sqrt%286%29%29%29+%2F%0D%0A+%28+sqrt%282%29%2Asqrt%282%29-sqrt%282%29sqrt%286%29%2Bsqrt%286%29sqrt%282%29-sqrt%286%29sqrt%286%29%29+=

+%284%28sqrt%282%29-sqrt%286%29%29%29+%2F%0D%0A+%28+sqrt%284%29-sqrt%2812%29%2Bsqrt%2812%29-sqrt%2836%29%29+=

+%284%28sqrt%282%29-sqrt%286%29%29%29+%2F%0D%0A+%28+sqrt%284%29-cross%28sqrt%2812%29%29%2Bcross%28sqrt%2812%29%29-sqrt%2836%29%29+=

+%284%28sqrt%282%29-sqrt%286%29%29%29+%2F%0D%0A+%282-6%29+=

+%284%28sqrt%282%29-sqrt%286%29%29%29+%2F%0D%0A+%28-4%29+=
Move the negative sign out front, and cancel the 4's

+-%28++cross%284%29%28sqrt%282%29-sqrt%286%29%29+%29+%2Fcross%284%29+=

-%28sqrt%282%29-sqrt%286%29%29

-sqrt%282%29%2Bsqrt%286%29

sqrt%286%29-sqrt%282%29

Edwin