FIND THE EXACT VALUE OF COS U 4/5, PI/2 <U<
SIN 2U=?
COS 2 U=
TAN 2 U=
To find the values of trigonometric functions of U, where U is an angle satisfying π/2 < U < π, given that cos(U) = 4/5, we can use the Pythagorean identity.
First, we can use the Pythagorean identity to find the value of sin(U):
sin^2(U) + cos^2(U) = 1
Substituting the value of cos(U) = 4/5:
sin^2(U) + (4/5)^2 = 1
sin^2(U) + 16/25 = 1
sin^2(U) = 1 - 16/25
sin^2(U) = 25/25 - 16/25
sin^2(U) = 9/25
Taking the square root of both sides:
sin(U) = ±√(9/25)
Since U is an angle in the second quadrant (π/2 < U < π), the value of sin(U) is positive. Therefore:
sin(U) = √(9/25) = 3/5
Now, we can use the double-angle identities to find the values of cos(2U) and tan(2U).
cos(2U) = cos^2(U) - sin^2(U)
Substituting the values of sin(U) = 3/5 and cos(U) = 4/5:
cos(2U) = (4/5)^2 - (3/5)^2
cos(2U) = 16/25 - 9/25
cos(2U) = 7/25
tan(2U) = 2tan(U) / (1 - tan^2(U))
Substituting the values of sin(U) = 3/5 and cos(U) = 4/5:
tan(2U) = 2(3/5) / (1 - (3/5)^2)
tan(2U) = 6/5 / (1 - 9/25)
tan(2U) = 6/5 / (16/25)
tan(2U) = (6/5) * (25/16)
tan(2U) = 15/8