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17 problems match “trigonometric identities”. Clear

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11. Minimum of a trigonometric sum ●●●○○ 50 ptsFor real numbers $x$ for which all six functions are defined, find the minimum value of $\left|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x\right|$. Round your answer to two decimal places.
12. Area swept by a rotating cardioid ●●●○○ 50 ptsThe polar curve $r = 1 + \cos \theta$ is rotated once around the point with polar coordinates $(2,0)$. What is the area of the region it sweeps over? Round to two decimal places.
13. Minimum of a product of trigonometric factors ●●●○○ 50 ptsFind the minimum value of the function $y=(4-3\sin x)(4-3\cos x)$ over real $x$. Give your answer as a decimal.
14. Maximum of tan beta under a constraint ●●●○○ 50 ptsGiven $\alpha, \beta \in\left(0, \frac{\pi}{2}\right)$ with $\alpha+\beta \neq \frac{\pi}{2}$ and $\sin \beta=2 \cos (\alpha+\beta) \sin \alpha$, find the maximum value of $\tan \beta$. Round to two decimal places.
15. Sum of four inverse trigonometric terms ●●●○○ 50 ptsFind the value, in degrees, of $\arcsin \frac{1}{\sqrt{10}}+\arccos \frac{5}{\sqrt{26}}+\arctan \frac{1}{7}+\arccos \frac{8}{\sqrt{65}}$.
16. Sine of a shifted double angle ●●●○○ 50 ptsGiven $f(x)=2\cos \left(x-\frac{\pi}{3}\right)\cos x+1$, and $f\left(\alpha +\frac{\pi}{12}\right)=\frac{7}{6}$ with $\alpha \in \left(0,\frac{\pi}{2}\right)$, find $\sin \left(\frac{7\pi}{6}-2\alpha \right)$. Round to two decimal places.
17. Sine of a difference from given values ●●●○○ 50 ptsGiven that $\alpha, \beta \in \left(\frac{\pi}{3}, \frac{5\pi}{6}\right)$, if $\sin\left(\alpha + \frac{\pi}{6}\right) = \frac{4}{5}$ and $\cos\left(\beta - \frac{5\pi}{6}\right) = \frac{5}{13}$, find the value of $\sin(\alpha - \beta)$. Round to two decimal places.

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