Thread: Double-Angle Formulas

Find the exact values of sin 2u, cos 2u, and tan 2u using the double angle formulas.:
cos u = -2/3, pi/2 < u < pi

So I know that the formulas are:
sin 2u = 2 sin u cos u
cos 2u = cos^2 u - sin^2 u
tan 2u = (2 tan u) / (1 - tan^2 u)

Now I am having trouble figuring out what sin u is. I believe that it is
sqrt(3), but I am not sure.
If I am right, the first formula would be:
sin 2u = (2) (sqrt3) (-2/3)
but I'm not sure how to multiply a square root with a fraction.

Any help is appreciated.

Originally Posted by iluvmathbutitshard

Find the exact values of sin 2u, cos 2u, and tan 2u using the double angle formulas.:
cos u = -2/3, pi/2 < u < pi

So I know that the formulas are:
sin 2u = 2 sin u cos u
cos 2u = cos^2 u - sin^2 u
tan 2u = (2 tan u) / (1 - tan^2 u)

Now I am having trouble figuring out what sin u is. I believe that it is
sqrt(3), but I am not sure.
If I am right, the first formula would be:
sin 2u = (2) (sqrt3) (-2/3)
but I'm not sure how to multiply a square root with a fraction.

Any help is appreciated.

nope, that's not the right value for sin(u). there are several ways you can get to sin(u). Two conventional ways are:

Use the fact that

Solve for . Note that we are in the second quadrant so you will need to take the positive square root. As sine is positive in that quadrant.


Another way: Draw a right-triangle with an acute angle . The adjacent side to should be labeled while the hypotenuse should be labeled . Use Pythagoras' Theorem to find the opposite side and use sine = opposite/hypotenuse

So sin u would be sqrt5/3?

If so, how would I multiply this by -2/3?

(Thank you for your help)

Originally Posted by iluvmathbutitshard

So sin u would be sqrt5/3?

yes,

If so, how would I multiply this by -2/3?

(Thank you for your help)

the same way you multiply any two fractions... multiply the numerators together to get the new numerator and the denominators together to get the new denominator.

Ok, great

SO then for cos 2u:
cos^2 u - sin^2 u
(4/9) - (5/9) = (-1/9)

Is this correct?

Originally Posted by iluvmathbutitshard

Ok, great

is it really? What was your answer for sin(2u)?

SO then for cos 2u:
cos^2 u - sin^2 u
(4/9) - (5/9) = (-1/9)

Is this correct?

yup

My answer for sin u was
(-4sqrt5) / (9)

And one last thing, is my tan correct:
(2 tan u) / (1 - tan^2 u)
(2 (-sqrt5/2)) / (1 - (5/4))

So this would simplify, I think, to
[(-2 sqrt5) / (2)] / [-1/4]
Is this correct?

Originally Posted by iluvmathbutitshard

My answer for sin u was
(-4sqrt5) / (9)

ok, good.

And one last thing, is my tan correct:
(2 tan u) / (1 - tan^2 u)
(2 (-sqrt5/2)) / (1 - (5/4))

So this would simplify, I think, to
[(-2 sqrt5) / (2)] / [-1/4]
Is this correct?

that's not simplified

for this last question, I would use the fact that

just divide the first two answers you got...

So the answer would be 4 sqrt5?

Originally Posted by iluvmathbutitshard

So the answer would be 4 sqrt5?

yup

Thank you so much for your help.