SOLUTION: Find the exact value of the expression: sin(4pi/9)cos(pi/9) - cos(4pi/9)sin(pi/9)

Algebra ->  Trigonometry-basics -> SOLUTION: Find the exact value of the expression: sin(4pi/9)cos(pi/9) - cos(4pi/9)sin(pi/9)      Log On


   

Question 1137318: Find the exact value of the expression:
sin(4pi/9)cos(pi/9) - cos(4pi/9)sin(pi/9)
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

sin%284pi%2F9%29cos%28pi%2F9%29+-+cos%284pi%2F9%29sin%28pi%2F9%29+

Use the following identity :
sin+%28s%29cos+%28t%29-cos+%28s%29sin+%28t%29=sin+%28s-t%29+

in your case s=4pi%2F9+and t=pi%2F9



sin%284pi%2F9%29cos%28pi%2F9%29+-+cos%284pi%2F9%29sin%28pi%2F9%29+=sin+%283pi%2F9++%29+

sin%284pi%2F9%29cos%28pi%2F9%29+-+cos%284pi%2F9%29sin%28pi%2F9%29+=sin+%28pi%2F3++%29+


Use the following identity : sin++%28+pi%2F3+%29=+sqrt%283%29%2F2

sin%284pi%2F9%29cos%28pi%2F9%29+-+cos%284pi%2F9%29sin%28pi%2F9%29+=sqrt%283%29%2F2




Answer by ikleyn(52797) About Me  (Show Source):
You can put this solution on YOUR website!
.
Apply the formula for the sine of the arguments' difference 


sin(a-b) = sin(a)*cos(b) - cos(a)*sin(b).


You will get


sin%284pi%2F9%29cos%28pi%2F9%29+-+cos%284pi%2F9%29sin%28pi%2F9%29 = sin%284pi%2F9-pi%2F9%29 = sin%283pi%2F9%29 = sin%28pi%2F3%29 = sqrt%283%29%2F2.


Solved, answered and completed.