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Inverse trig equations

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  1. Sum of inverse cosine and sine ●○○○○ 10 ptsFind the value of cos⁡−1(12)+2sin⁡−1(12)\cos^{-1}\left(\frac{1}{2}\right) + 2\sin^{-1}\left(\frac{1}{2}\right). Give your answer in degrees.
  2. Find x+y from a trigonometric system ●●○○○ 20 ptsGiven that x+sin⁡y=2008x+\sin y=2008 and x+2008cos⁡y=2007x+2008 \cos y=2007, where 0≤y≤π20 \leq y \leq \frac{\pi}{2}, find the value of x+yx+y. Round to two decimal places.
  3. Sum of inverse sines of sines ●●○○○ 20 ptsCompute sin⁡−1(sin⁡3)+sin⁡−1(sin⁡4)+sin⁡−1(sin⁡5).\sin^{-1} (\sin 3) + \sin^{-1} (\sin 4) + \sin^{-1} (\sin 5). All functions are in radians.
  4. Cotangent of an inverse trigonometric sum ●●○○○ 20 ptsIf cot⁡(cos⁡−135+sin⁡−1x)=0\cot\left(\cos^{-1}\frac{3}{5}+\sin^{-1}x\right)=0, find the value of xx. Give your answer as a decimal.
  5. Sum of x and y in a system ●●●○○ 50 ptsGiven that x+sin⁡y=2008x+\sin y=2008 and x+2008cos⁡y=2007x+2008 \cos y=2007, where 0≤y≤π20 \leq y \leq \frac{\pi}{2}, find the value of x+yx+y. Give your answer as a decimal rounded to two decimal places.
  6. Square root plus arcsine value ●●●○○ 50 ptsGiven a real number xx, find the value of x(x+1)+arcsin⁡x2+x+1\sqrt{x(x+1)}+\arcsin \sqrt{x^{2}+x+1}, in radians. Round to two decimal places.
  7. Sum of four inverse trigonometric terms ●●●○○ 50 ptsFind the value, in degrees, of arcsin⁡110+arccos⁡526+arctan⁡17+arccos⁡865\arcsin \frac{1}{\sqrt{10}}+\arccos \frac{5}{\sqrt{26}}+\arctan \frac{1}{7}+\arccos \frac{8}{\sqrt{65}}.

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