trig double angle formula?

Question

use the doble angle formula to determine the exact value of the sin(2u), cos(2u) and tan(2u) if tan u = 2 and cos u >0

Answer 1

this is just a try buddy!

the formula for tan (2u) = 2 tan u/(1 - tan^2u) now if tan u = 2

substitute the value to the formula

tan 2u = 2(2)/(1 - 4)

tan 2u = - 4/3

sin 2u = 2 (sin u) (cos u)

the value of sin u and cos u can be solve using the given tan u = 2 by refering this to the right triangle.

note that tan = opposite / adjacent

so sin u = 2/(sqrt of (5)) and cos u = 1/(sqrt of (5))

for sin 2u = 2 (sin u) (cos u) substitute

sin 2u = 2(2/(sqrt of (5)))(1/(sqrt of (5))

sin 2u = 4/5

for cos 2u = (cosu)^2 - (sinu)^2 substitue the value of sin u and cos u

cos 2u = [1/(sqrt of (5))]^2 - [2/(sqrt of (5))]^2

cos 2u = -3/5

Answer 2

Use the double angle formula to determine the exact value of the sin(2u), cos(2u) and tan(2u)

If tan u = 2 and cos u > 0

These two conditions mean u is in the first quadrant. Since tan u >1, u > π/4 and 2u > π/2. So 2u is in the second quadrant.

tan(2u) = 2tanu / (1 - tan²u) = 2*2 / (1 - 2²2) = 4/(1 - 4) = -4/3

tan(2u) = sin(2u) / cos(2u) = -4/3 = 4/-3 = (4/x) / (-3/x)

sin²(2u) + cos²(2u) = (4/x)² + (-3/x)² = 16/x² + 9/x² = 1

16 + 9 = 25 = x²
x = 5

tan(2u) = -4/3
sin(2u) = 4/x = 4/5
cos(2u) = -3/x = -3/5

Answer 3

Hopefully you know the double angle formula and can apply it once you figure out what sin u and cos u are. To do this, rewrite tan u = 2 as (sin u) / (cos u) = 2, so that sin u = 2 cos u. Then recall that sin^2 u + cos^2 u =1. By combining the last two equations you can solve for cos u, then get sin u, and then apply the double angle formulas.

Answer 4

Draw a triangle with 2 as the vertical leg and 1 as the horizontal leg. The lower acute angle would be "u" such that tan(u) = 2. Find the hypoteneuse: sqrt(5). So sin(u) = 2/sqrt(5) while cos(u) = 1/sqrt(5). (Don't bother to rationalize denominators yet.)

Plug into the various formulas. You can surely handle that!