Question

Question #b3353

Answer 1

tan 2u = 240/161.

Explanation:

Think of a right triangle with sides 8, 15, 17 --

`8^{2}+15^{2} =17^{2}`.

The angle at the hypoteneuse and the longer leg has a cosine of 15/17 and a sine of 8/17. Call that angle w.

Adding `\pi` to w gives angle u between `\pi` and `3\pi/2`, with cos u = -cos w = -15/17. Then sin u = -sin w = -8/17. Dividing the sine by the cosine gives tan u = 8/15.

Now use the double angle formula,

tan 2u
= 2tan u/(1-`tan^{2} u`)

We have tan u = 8/15. So tan 2u = (16/15)/(1-64/225) = 240/161.

Answer 2

`tan 2u = 240/161`

Explanation:

`cos u = -15/17`
`sin^2 u = 1 - 225/289 = +- 64/17`
Since `pi < u < (3pi)/2` , then `sin u = -8/17`
`tan u = (sin u)/(cos u) = (-8/17)(-17/15) = 8/15`
To find tan 2u apply the trig identity:
`tan 2u = (2tan u)/(1 - tan^2 u)`
Proceed numerically:
`2tan u = 2(8/15) = 16/15`
`(1 - tan^2 u) = 1 - 64/225 = 161/225`
Answer: `tan 2u = (16/15)(225/161) = 240/161`