Question 910327: If 7sin square x + 3cos square x = 4 then find value of tanx
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! looks like your angle is going to be 30 degrees.
here's why:
start with 7 * sin^2(x) + 3 * cos^2(x) = 4
subtract cos^2(x) from both sides of the equation to get:
7 * sin^2(x) = -3 * cos^2(x) + 4
divide both sides of the equation by sin^2(x) to get:
7 * sin^2(x) / sin^2(x) = -3 * cos^2(x) / sin^2(x) + 4 / sin^2(x)
sin^2(x) / sin^2(x) = 1
cos^2(x) / sin^2(x) = cot^2(x)
1 / sin^2(x) = csc^2(x)
your equation becomes:
7 = -3 * cot^2(x) + 4 * csc^2(x)
since csc^2(x) is equal to 1 + cot^2(x), your equation becomes:
7 = -3 * cot^2(x) + 4 * (1 + cot^2(x))
simplify to get:
7 = -3 * cot^2(x) + 4 + 4 * cot^2(x)
subtract 4 from both sides of the equation to get:
3 = -3 * cot^2(x) + 4 * cot^2(x)
combine like terms to get:
3 = cot^2(x)
take the square root of both sides of the equation to get:
+/- sqrt(3) = cot(x)
you get:
cot(x) = sqrt(3) or cot(x) = -3
since tan(x) is equal to 1/cot(x), then you get:
1/tan(x) = sqrt(3) or 1/tan(x) = -sqrt(3)
solve for tan(x) and you get:
tan(x) = 1/sqrt(3) or tan(x) = -1/sqrt(3)
your solution is:
tan(x) = plus or minus 1 / sqrt(3)
if you need to rationalize the denominator, then your solution is:
tan(x) = plus or minus sqrt(3)/3
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