If #cos(theta) = (x-1)/(x+1)#, what is sin and tan of theta?

1 Answer
Jim G.
May 22, 2016

#sintheta=(2sqrtx)/(x+1), tantheta=(2sqrtx)/(x-1)#

Explanation:

Here is a sketch to clarify.

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Given #costheta=(x-1)/(x+1)=a/h#

We can identify the adjacent side (x-1) and the hypotenuse (x+1)

To find the opposite side (o) use #color(blue)"Pythagoras' theorem"#

For this right triangle then.

#(x+1)^2=(x-1)^2+o^2#

expand using FOIL and collect like terms

#rArrx^2+2x+1=x^2-2x+1+o^2#

#rArro^2=cancel(x^2)-cancel(x^2)+2x+2xcancel(+1)cancel(-1)=4x#

now #o^2=4xrArro=sqrt(4x)=2sqrtx#

#rArrsintheta=(2sqrtx)/(x+1)" and " tantheta=(2sqrtx)/(x-1)#

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