Generating...                               s20quiz10_n10

  1. Choose the correct trigonometric identity.

    $\sin( t-{{3\,\pi}\over{2}}) = -\sin t$

    $\sin( t-{{3\,\pi}\over{2}}) = \cos t$

    $\sin( t-{{3\,\pi}\over{2}}) = \sin t$

    $\sin( t-{{3\,\pi}\over{2}}) = -\cos t$

  2. Choose the correct trigonometric identity.

    $\displaystyle -{{\sin \left(2\,t\right)}\over{1-\cos \left(2\,t\right)}}=-\tan t$

    $\displaystyle -{{\sin \left(2\,t\right)}\over{1-\cos \left(2\,t\right)}}=-\cot t$

    $\displaystyle -{{\sin \left(2\,t\right)}\over{1-\cos \left(2\,t\right)}}=\cot t$

    $\displaystyle -{{\sin \left(2\,t\right)}\over{1-\cos \left(2\,t\right)}}=\tan t$

  3. Find the derivative $f'(x)$ for $f(x) =\displaystyle e^{x}\,\sec x $ .

    $f'(x) =\displaystyle e^{x}\,\left(\cot x-1\right)\,\csc x $ $f'(x) =\displaystyle e^{x}\,\sec x\,\left(\tan x+1\right) $ $f'(x) =\displaystyle -e^{x}\,\sec x\,\left(\tan x+1\right) $ $f'(x) =\displaystyle -e^{x}\,\left(\cot x-1\right)\,\csc x $

  4. Suppose that $0 < t < \frac{\pi}{2}$, and that $\displaystyle \csc t=2$ . Find the values of the trigonometric functions $\sin(t)$, $\cos(t)$, and $\tan(t)$.

    $\displaystyle \left[ \sin t={{\sqrt{3}}\over{2}} , \cos t={{1}\over{2}} , \tan t= \sqrt{3} \right] $

    $\displaystyle \left[ \sin t={{1}\over{2}} , \cos t={{\sqrt{3}}\over{2}} , \tan t= {{1}\over{\sqrt{3}}} \right] $

    $\displaystyle \left[ \sin t={{1}\over{2}} , \cos t=-{{1}\over{3}} , \tan t=-{{3 }\over{2}} \right] $

    $\displaystyle \left[ \sin t=-{{1}\over{3}} , \cos t={{1}\over{2}} , \tan t=-{{2 }\over{3}} \right] $

  5. Find the derivative $f'(x)$ for $f(x) =\displaystyle \cot x\,\sec x $ .

    $f'(x) =\displaystyle \sec x $ $f'(x) =\displaystyle \sec x\,\tan x $ $f'(x) =\displaystyle -\cot x\,\csc x $ $f'(x) =\displaystyle \cot x\,\csc x $ $f'(x) =\displaystyle -\sec x\,\tan x $

  6. Find the derivative $f'(x)$ for $f(x) =\displaystyle {{\cos x}\over{x^3}} $ .

    $f'(x) =\displaystyle {{3\,\sin x-x\,\cos x}\over{x^4}} $ $f'(x) =\displaystyle -{{x\,\sin x+3\,\cos x}\over{x^4}} $ $f'(x) =\displaystyle -{{3\,\sin x-x\,\cos x}\over{x^4}} $ $f'(x) =\displaystyle {{x\,\sin x+3\,\cos x}\over{x^4}} $

  7. Suppose that $\frac{\pi}{2} < t < \pi$, and that $\displaystyle \csc t=4$ . Find the values of the trigonometric functions $\sin(t)$, $\cos(t)$, and $\tan(t)$.

    $\displaystyle \left[ \sin t={{1}\over{4}} , \cos t=-{{\sqrt{15}}\over{4}} , \tan t=-{{1}\over{\sqrt{15}}} \right] $

    $\displaystyle \left[ \sin t=-{{\sqrt{15}}\over{4}} , \cos t={{1}\over{4}} , \tan t=-\sqrt{15} \right] $

    $\displaystyle \left[ \sin t={{\sqrt{15}}\over{4}} , \cos t={{1}\over{4}} , \tan t= \sqrt{15} \right] $

    $\displaystyle \left[ \sin t={{1}\over{4}} , \cos t={{\sqrt{15}}\over{4}} , \tan t= {{1}\over{\sqrt{15}}} \right] $

  8. Find the derivative $f'(x)$ for $f(x) =\displaystyle -2\,\cos x $ .

    $f'(x) =\displaystyle -2\,\sin x $ $f'(x) =\displaystyle -2\,\cos x $ $f'(x) =\displaystyle -2\,\sin x $ $f'(x) =\displaystyle 2\,\sin x $ $f'(x) =\displaystyle 2\,\cos x $

  9. Choose the correct trigonometric identity.

    $\displaystyle \sin x-\cos x\,\cot x=\csc x$

    $\displaystyle \sin x-\cos x\,\cot x=-\csc x$

    $\displaystyle \sin x-\cos x\,\cot x={{2\,\csc x}\over{\left(\sec x\right)^2}}- \csc x$

    $\displaystyle \sin x-\cos x\,\cot x=2\,\sin x-\csc x$

  10. Suppose that $\displaystyle g\left( -{{\pi}\over{4}} \right) = 3 $ and $\displaystyle g'\left( -{{\pi}\over{4}} \right) = 0 $. Then find the derivative $\displaystyle f'\left( -{{\pi}\over{4}} \right)$ for $f(x) = \displaystyle g\left(x\right)\,\cos x $ .

    0 $\displaystyle {{3}\over{\sqrt{2}}}$ $\displaystyle {{3}\over{\sqrt{2}}}$ $\displaystyle -{{3}\over{\sqrt{2}}}$ $\displaystyle {{3}\over{\sqrt{2}}}$


Department of Mathematics
Last modified: 2025-10-30