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

Thanks!

2 Answers
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.

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