Question 180061
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Trigonometry-basics/180061: Determine exact values.
Find sin(A) and cos(A) if tan(A) = 5/3 and cos(A) < 0.
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Calculations in the post by @HyperBrain are incorrect, producing wrong answer.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;For correct solution, see my post below.



Let A be an angle, sin(A) = a/c, cos(A) = b/c, tan(A) = a/b. Where
a=opposite side
b=adjacent side
c=hypotenuse

a, b, and c form a right triangle and therefore must satisfy,

{{{a^2+b^2=c^2}}}


Since tan(A) is positive and cos(A) is negative (given),
we conclude from it that A is in QIII.


Since tan(A) = {{{5/3}}}, let a = -5, b = -3.


{{{c^2}}} = {{{5^2+3^2}}} = 25+9 = 34,
c = {{{sqrt(34)}}}


sin(A) = {{{-5/sqrt(34)}}} = {{{-(5sqrt(34))/34}}}


cos(A) = {{{-3/sqrt(34)}}} = {{{-(3sqrt(34))/34}}}


Solved correctly.