Answer to Question #243483 in Calculus for Anand

Question #243483
find the maximum height of the curve
y = 4sin x - 3cos x above the x axis.
1
Expert's answer
2021-09-29T00:29:06-0400

Find the maximum height of the curve "y = 4\\sin x - 3\\cos x" above the "x" axis.


"\\sqrt{(4)^2+(3)^2}=5"

"y =5( \\dfrac{4}{5}\\sin x - \\dfrac{3}{5}\\cos x)"

Let "\\cos \\theta=\\dfrac{4}{5}, \\sin\\theta=\\dfrac{3}{5}, \\theta=\\sin^{-1}(\\dfrac{3}{5})."

Then

"\\dfrac{4}{5}\\sin x - \\dfrac{3}{5}\\cos x=\\sin(x-\\theta), \\theta=\\sin^{-1}(\\dfrac{3}{5})"

"y=5\\sin^{-1}(\\dfrac{3}{5})"

"-1\\leq\\sin^{-1}(\\dfrac{3}{5})\\leq 1"

"-5\\leq y\\leq 5"

The maximum height of the curve of the curve "y = 4\\sin x - 3\\cos x" above the "x" axis is "5."



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