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

1 Answer
Noah G
Apr 16, 2016

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.

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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

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