How do you find cos if sin= 5/13?

2 Answers
Alan P.
Nov 20, 2015

#cos=+-12/13#

Explanation:

If #sin=5/13#
then
#color(white)("XXX")# the ratio of #("opposite side")/("hypotenuse") = 5/13#

By Pythagorean Theorem
If #"opposite side" = 5 " units"# and #"hypotenuse" = 13 " units"#
#color(white)("XXX")#(for any #"units"#)
then #"adjacent side" = 12 " units"#
and
#cos = ("adjacent side")/("hypotenuse") = 12/13#

However, we need to note that if the angle is in Quadrant II then the #"adjacent side"# will actually be a negative value,
so
#color(white)("XXX")cos=12/13# for an angle in Q I
or
#color(white)("XXX")cos=-12/13# for an angle in Q II
(the angle can't be in Q III or Q IV, since #sin > 0#)

Nghi N
May 21, 2017

#cos x = +- 12/13#

Explanation:

Another way.
#cos^2 x = 1 - sin^2 x = 1 - 25/169 = 144/169#
#cos x = +- 12/13#
sin x > 0 --> x is either in Quadrant 1 or Quadrant 2.

If x is in Quadrant 1, then, cos x > 0 --> #x = 12/13#
If x is in Quadrant 2, then, cos < 0 --> #cos x = - 12/13#

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