Question

`csc lambda = 17/8` in quadrant 2. What is `tan 2lambda`?

Answer 1

First of all, we must remember that `csc = 1/sin`

Explanation:

Thus, we can conclude that `sinlambda = 8/17`

Since sin is opposite/hypotenuse, we must find the adjacent side, because tan is opposite/adjacent. We can do this by pythagorean theorem. The following diagram shows how we can draw a right triangle on the cartesian plane to find the value of the six trigonometric ratios.

Rearranging our pythagorean theorem to find adjacent side `b`, we get:

`b^2 = c^2 - a^2`

`b^2 = 17^2 - 8^2`

`b^2 = 225`

`b = 15`

Therefore, the adjacent side from `lambda` measures -15 units (the x axis is negative in quadrant II). We can conclude that `tanlambda = -8/15`

Now, since it's `2lambda` that we must find, we need to use the double angle formula for `tan`: `tan(2lambda) = (2tanlamda)/(1 - tan^2lamda`

Substituting:

`tan(2lamda) = (2(-8/15))/(1 - (-8/15)^2)`

`tan(2lambda) = (-16/15)/(161/225)`

`tna(2lamda) = -16/15 xx 225/161`

`tan(2lambda) = -240/161`

Hopefully this helps