Find sin 2x, cos 2x, and tan 2x from the given information: sin x = -5/13 , x in Quadrant III, sin 2x = cos 2x = tan 2x =

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Answer 1

The reason for telling you the quadrant is to make sure that you get the correct sign for the cos x function.  Find the value of cos x using the following:

 

sin²(x) + cos²(x) = 1;  or cos²(x) = 1-sin²(x)

 

cos²(x) = 1-(-5/13)² => cos²(x) = 144/169;  

 

In quadrant III both the sin and cos are negative so

cos(x) = -12/13 (after taking square roots).

 

Then tan(x) = sin(x)/cos(x) = (-5/13)/(-12/13) = 5/12.

 

Now you can use the angle addition formulas to find sin(2x), cos(2x), and tan(2x).

 

sin(x + x) = sinx * cosx + cosx * sinx 

               = (-5/13)*(-12/13) + (-12/13)(-5/13) = 120/169

 

cos(x + x) = cosX * cos(x) - sinx*sinx 

                = (-12/13)(-12/13) - (-5/13)(-5/13)

                =  119/169

 

You could use the tan double angle formula, but it is easiest to use 

 

tan(2x) = sin(2x)/cos(2x)  = (120/169) / (119/169)

           = 120/119.