Expert: Sherman D. - 8/12/2006

Question
x is an angle.
given sin x = 6/7, and 1/2 pi < x < pi, find the exact values of cos x, tan x, sin 2x, cos 2x and
tan 2x.

using pythagoras theorm i worked out the other side
7^2-6^2=root 13.
cos x = (root13)/7, but because the area is 1/2 pi to pi (quadrant where only sin is positive), its cos x = -(root13)/7; same with
tan x= -6/root13
(correct me if im wrong)
With sin 2x, the angle is multipled by 2, so im dividing; (6/7) * 1/2 = 6/14. Is my logic correct?
same goes for Cos 2x = -(root13)/14;
tan 2x = -6/root13.

Thankyou. for your time

Answer
sinA = y/r
cosA = x/r
tanA = y/x

sinA = 6/7

x^2 + y^2 = r^2
x^2 + 6^2 = 7^2
x^2 + 36 = 49
x^2 = 13
x = sqrt(13)

cosA = (-sqrt(13))/7
tanA = 6/(sqrt(13)) = (-6sqrt(13))/13

So for you are correct.

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Here is the tricky part

sin(2A) = 2cosA * sinA
cos(2A) = -(sinA)^2 + (cosA)^2
tan(2A) = (2tanA)/(1 - (tanA)^2)

since you already know cosA, sinA, tanA, this should be hard

sin(2A) = 2cosA * sinA
sin(2A) = 2(-sqrt(13)/7) = (-2sqrt(13))/7

cos(2A) = -(sinA)^2 + (cosA)^2
cos(2A) = -(6/7)^2 + ((-sqrt(13))/7)^2
cos(2A) = -(36/49) + (13/49)
cos(2A) = (-36 + 13)/49
cos(2A) = (-23/49)

tan(2A) = (2tanA)/(1 - (tanA)^2)
tan(2A) = (2(-6sqrt(13))/13)/(1 - ((-6sqrt(13))/13)^2)
tan(2A) = ((-12sqrt(13))/13)/(1 - (468/169))
tan(2A) = ((-12sqrt(13))/13)/(1 - (36/13))
tan(2A) = ((-12sqrt(13))/13)/((13/13) - (36/13))
tan(2A) = ((-12sqrt(13))/13)/((13 - 36)/13)
tan(2A) = ((-12sqrt(13))/13)/(-23/13)
tan(2A) = ((-12sqrt(13))/13) * (-13/23)
tan(2A) = ((-12 * -13 * sqrt(13))/(13 * 23)
tan(2A) = (12sqrt(13))/23

Just try it, and i can guarantee they will come out the same. I used the sin(x) = 6/7 and therefore x = sin^-1(6/7) method to find out if it works out or not. When you do, you will get -1 if you divide both sides, but because of your circumstances, you will get a + answer and not a - answer.


This shows you how to get cos(x), sin(x), tan(x), etc... with what you have.
http://id.mind.net/~zona/mmts/trigonometryRealms/TrigFuncPointDef/TrigFuncPointD...

and

This site will show you how to work sin(2x), cos(2x), tan(2x) and more.
http://mcraeclan.com/MathHelp/GeometryTrigEquivCos3xEtc.htm