use the given information to find cos2t & sin2t. cost= 4/5, 0<t<pi/2

Question

use the given information to find cos2t & sin2t. cost= 4/5, 0<t<pi/2

PLEASE SHOW WORK & EXPLAIN

Answer 1

since t is in first quadrant, it's obvious that sin t = 3/5 (think 3-4-5 rt. triangle),

 

cos 2t = 2 cos²t - 1 = 2(16/25) - 1 = [32 - 25] / 25 = 7/25

 

sin 2t = 2 sin t cost = 2(4/5)(3/5) = 24/25.

 

Note that  the sum of the squares of these two results above does = 1 (just checking)

Answer 2

For cos2t, use the identities

 

cos(x + y) = cosx cosy - sinx siny

sin²x = 1 - cos²x

 

cos2t = cos(t + t) = cos²t - sin²t = (4/5)² - [1 - (4/5)²] = ?

 

For sin2t, use the identity

 

sin(x + y) = sinx cosy + cosx siny

 

sin2t = sin(t + t) = sint cost + cost sint = 2sint cost

 

cost = 4/5

sint = √(1 - cos²t) = +√[1 - (4/5)²]  (use positive square root because 0 < t < π/2)

 

sin2t = 2(4/5)√[1 - (4/5)²] = ?