Find the exact value of the expression.
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Answer:  

475/493

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sine subtraction identity can be applied here:

   

we know that

   

we need the values of 

   



recall that for a point  on the terminal arm of angle  in standard pos, we have, by definition:

   

where  is the distance from the origin (0,0) to the point  . (r is never negative.) this would be the hypotenuse length when forming a reference triangle. 

focusing on angle  , refer to figure 1. 
it is an angle in quadrant 2.

since 

   

and we want to find , we need to find the x-value to get the exact value of cosine of alpha.

the y-coordinate is 21 and distance from the point to the origin is 29. a reference right-triangle is formed and we can calculate the corresponding x-coordinate using the pythagorean theorem.

the x-coordinate turns out to be  , as shown.

so

   


focusing on angle  , refer to figure 2

we're given

   

we want to find  , defined as 
   
the x-coordinate is 15 and distance from the point to the origin is 17. a reference right-triangle is formed and we can calculate the corresponding y-coordinate using the pythagorean theorem.

y-coordinate is 8.

so

   



our info:

   

calculate using the subtraction identity (: