If sin x = -12/13 and tan x is positive, find the values of cos x and tan x ?

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2 Answers
Max G.
Feb 6, 2018

Determine the Quadrant first

Explanation:

Since #tanx > 0#, the angle is in either Quadrant I or Quadrant III.
Since #sinx < 0#, the angle must be in Quadrant III.
In Quadrant III, cosine is also negative.

Draw a triangle in Quadrant III as indicated. Since #sin = (OPPOSITE)/(HYPOTENUSE)#, let 13 indicate the hypotenuse, and let -12 indicate the side that is opposite to angle #x#.

By the Pythagorean Theorem, the length of the adjacent side is
#sqrt(13^2 - (-12)^2) = 5#.
However, since we are in Quadrant III, the 5 is negative. Write -5.

Now use the fact that #cos = (ADJACENT)/(HYPOTENUSE)#
and #tan = (OPPOSITE)/(ADJACENT)# to find the values of the trig functions.

Jim G.
Feb 6, 2018

#cosx=-5/13" and "tanx=12/5#

Explanation:

#"using the "color(blue)"trigonometric identity"#

#•color(white)(x)sin^2x+cos^2x=1#

#rArrcosx=+-sqrt(1-sin^2x)#

#"since "sinx<0" and "tanx>0#

#"then x is in the third quadrant where "cosx<0#

#rArrcosx=-sqrt(1-(-12/13)^2)#

#color(white)(rArrcosx)=-sqrt(25/169)=-5/13#

#tanx=sinx/cosx=(-12/13)/(-5/13)=-12/13xx-13/5=12/5#

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