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1. find the exact value of tan(15) using sum or difference identity

2. find the exact value of cos2x when sin(x) = 3/5 and pi/2 < x < pi. 

 Dec 5, 2018
 #1
avatar+17332 
+2

  tan (15) = sin/cos         sin (45-30) = sin45 cos 30 - cos 45 sin30     =

                                  cos(45-30) =  cos45cos30 + sin 45 sin 30        

 

    sqrt2/2 sqrt3/2   -  sqrt2/2  1/2   =   sqrt2/2 (sqrt3/2-1/2)        =        1/2 (sqrt3-1)        =   sqrt3-1

    sqrt2/2 sqrt3/2  +   sqrt2/2  1/2       sqrt2/2 (sqrt3/2 + 1/2)               1/2 (sqrt3+1)            sqrt3+1

 

Multiply num and denom by  (sqrt3-1)     to finally get    tan(15) = (2- sqrt3)

 Dec 6, 2018
 #2
avatar+17332 
+1

2

 

cos(2x) = 1-2 sin^2 x  =  1- 2(3/5)^2 = 1 - 18/25 = 7/25

 Dec 6, 2018

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