Mathegoras

Trigonometry › Trigonometric functions: period and extremes

Maximum and minimum

Solve the first problem 17 problems

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  1. Extrema of sine on an interval ●○○○○ 10 ptsFind the xx-values in [0,2π][0, 2\pi] at which f(x)=sin⁡xf(x)=\sin x attains its maximum and its minimum. Give each in degrees, with the maximum first and then the minimum.
  2. Maximum of a trigonometric quadratic in cosine ●●○○○ 20 ptsFind the maximum value of the function f(x)=sin⁡2x+3cos⁡x−34f(x)=\sin^{2}x+\sqrt{3}\cos x-\frac{3}{4} for x∈[0,π2]x\in\left[0,\frac{\pi}{2}\right].
  3. Maximum of x+y under a trig constraint ●●○○○ 20 ptsFind the maximum value of x+yx+y if (2sin⁡x−1)(2cos⁡y−3)=0(2\sin x-1)(2\cos y-\sqrt{3})=0, x∈[0,3π/2]x\in[0,3\pi/2], y∈[π,2π]y\in[\pi,2\pi]. Round to two decimal places.
  4. Max-min difference of a shifted sine ●●○○○ 20 ptsFind the difference between the maximum and minimum values of the function y=2sin⁡(πx6−π3)y=2\sin\left(\frac{\pi x}{6}-\frac{\pi}{3}\right) for 0⩽x⩽90\leqslant x\leqslant 9. Round to two decimal places.
  5. Minimum of a trigonometric sum ●●●○○ 50 ptsFor real numbers xx for which all six functions are defined, find the minimum value of ∣sin⁡x+cos⁡x+tan⁡x+cot⁡x+sec⁡x+csc⁡x∣\left|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x\right|. Round your answer to two decimal places.
  6. Minimum of a product of trigonometric factors ●●●○○ 50 ptsFind the minimum value of the function y=(4−3sin⁡x)(4−3cos⁡x)y=(4-3\sin x)(4-3\cos x) over real xx. Give your answer as a decimal.
  7. Maximum of tan beta under a constraint ●●●○○ 50 ptsGiven α,β∈(0,π2)\alpha, \beta \in\left(0, \frac{\pi}{2}\right) with α+β≠π2\alpha+\beta \neq \frac{\pi}{2} and sin⁡β=2cos⁡(α+β)sin⁡α\sin \beta=2 \cos (\alpha+\beta) \sin \alpha, find the maximum value of tan⁡β\tan \beta. Round to two decimal places.

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