SOLUTION: Determine exact values. Find sin A and cos A if tan A=5/3 and cos A <0.

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Question 180061: Determine exact values.
Find sin A and cos A if tan A=5/3 and cos A <0.
Found 2 solutions by HyperBrain, ikleyn:
Answer by HyperBrain(694) About Me  (Show Source):
You can put this solution on YOUR website!
Let A be an angle, sin A=a/c, cos A=b/c, tan=a/b. Where
a=opposite side
b=adjacent side
c=hypotenuse
a, b, and c form a right triangle and therefore must satisfy,
a%5E2%2Bb%5E2=c%5E2
if tan A=5/3, let a=5, b=3
c%5E2=5%5E2%2B3%5E2=25%2B9=34
c=sqrt%2834%29
sin A=5%2Fsqrt%2834%29=%285sqrt%2834%29%29%2F34
cos A=3%2Fsqrt%2834%29=%283sqrt%2834%29%29%2F34
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Answer by ikleyn(53937) About Me  (Show Source):
You can put this solution on YOUR website!
.
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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        Calculations in the post by @HyperBrain are incorrect, producing wrong answer.

        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%5E2%2Bb%5E2=c%5E2

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

Since tan(A) = 5%2F3, let a = -5, b = -3.

c%5E2 = 5%5E2%2B3%5E2 = 25+9 = 34,
c = sqrt%2834%29

sin(A) = -5%2Fsqrt%2834%29 = -%285sqrt%2834%29%29%2F34

cos(A) = -3%2Fsqrt%2834%29 = -%283sqrt%2834%29%29%2F34

Solved correctly.