Question 166485
Use the angle sum or difference identity to find the exact value of
cos195 
<pre><font size = 4 color = "indigo"><b>
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(alpha+beta) = cos(alpha)cos(beta)-sin(alpha)sin(beta)}}}

{{{cos(195)=cos(150+45) = cos(150)cos(45)-sin(150)sin(45)}}}=

{{{(-sqrt(3)/2)(sqrt(2)/2)-(1/2)(sqrt(2)/2)}}}

{{{-sqrt(6)/4 -sqrt(2)/4}}}

{{{(-sqrt(6)-sqrt(2))/4}}}

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

{{{csc(5pi/12)}}}

Let's convert that to degrees:

{{{5pi/12=(5pi/12)(180/pi)=75}}}°

Then we can write 75° as 30°+45°

{{{csc(5pi/12)=csc(75)=1/sin(75)=1/sin(30+45)}}}=

Then we can use the identity: {{{sin(alpha+beta) = sin(alpha)cos(beta)+cos(alpha)sin(beta)}}}

{{{1/sin(30+45) = 1/(sin(30)cos(45)+cos(30)sin(45))}}}=

{{{1/((1/2)(sqrt(2)/2)+(sqrt(3)/2)(sqrt(2)/2))}}}=

{{{1/((sqrt(2)/4)+(sqrt(6)/4))}}}=

{{{1/(  (sqrt(2)+sqrt(6))/4  )}}}=

{{{matrix(1,3,     1, "÷", (sqrt(2)+sqrt(6))/4 )}}}

{{{matrix(1,3,     1, "×", 4/(sqrt(2)+sqrt(6)) )}}}

Rationalize the denominator:

{{{matrix(1,3,
 4/(sqrt(2)+sqrt(6))  ,

 "×",

(sqrt(2)-sqrt(6))/(sqrt(2)-sqrt(6))  

)}}}

Indicate multiplication of numerators and denominators:

{{{ (4(sqrt(2)-sqrt(6))) /
 ( sqrt(2)+sqrt(6) )( sqrt(2)-sqrt(6) ) }}}

Multiply out the bottom:

{{{ (4(sqrt(2)-sqrt(6))) /
 ( sqrt(2)*sqrt(2)-sqrt(2)sqrt(6)+sqrt(6)sqrt(2)-sqrt(6)sqrt(6)) }}}=

{{{ (4(sqrt(2)-sqrt(6))) /
 ( sqrt(4)-sqrt(12)+sqrt(12)-sqrt(36)) }}}=

{{{ (4(sqrt(2)-sqrt(6))) /
 ( sqrt(4)-cross(sqrt(12))+cross(sqrt(12))-sqrt(36)) }}}=

{{{ (4(sqrt(2)-sqrt(6))) /
 (2-6) }}}=

{{{ (4(sqrt(2)-sqrt(6))) /
 (-4) }}}=
Move the negative sign out front, and cancel the 4's

{{{ -(  cross(4)(sqrt(2)-sqrt(6)) ) /cross(4) }}}=

{{{-(sqrt(2)-sqrt(6))}}}

{{{-sqrt(2)+sqrt(6)}}}

{{{sqrt(6)-sqrt(2)}}}

Edwin</pre>