Generating...                               s20quiz10_n22

  1. Suppose that $\displaystyle g\left( {{\pi}\over{3}} \right) = 1 $ and $\displaystyle g'\left( {{\pi}\over{3}} \right) = 2 $. Then find the derivative $\displaystyle f'\left( {{\pi}\over{3}} \right)$ for $f(x) = \displaystyle g\left(x\right)\,\cos x $ .

    2 $\displaystyle -{{\sqrt{3}}\over{2}}-1$ $\displaystyle 1-{{\sqrt{3}}\over{2}}$ $\displaystyle {{\sqrt{3}}\over{2}}+1$ $\displaystyle {{1}\over{2}}$

  2. Choose the correct trigonometric identity.

    $\displaystyle \cos x-\sin x\,\tan x=\sec x$

    $\displaystyle \cos x-\sin x\,\tan x=2\,\cos x-\sec x$

    $\displaystyle \cos x-\sin x\,\tan x=-\sec x$

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

  3. Find the values of $\displaystyle \sin t$ and $\displaystyle \cos t$ for $\displaystyle t={{8\,\pi}\over{3}}$ .

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

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

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

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

  4. 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}\,\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) $

  5. Choose the correct trigonometric identity.

    $\displaystyle \left(\cot t\right)^2-\left(\csc t\right)^2=-1$

    $\displaystyle \left(\cot t\right)^2-\left(\csc t\right)^2={{\sin ^2t-2}\over{ \sin ^2t}}$

    $\displaystyle \left(\cot t\right)^2-\left(\csc t\right)^2=1$

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

  6. Choose the correct trigonometric identity.

    $\displaystyle \left(\sin ^2t+1\right)\,\left(\tan ^2t+1\right)=-{{2\,\sin ^4t+ \sin ^2t-1}\over{\cos ^2t}}$

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

    $\displaystyle \left(\sin ^2t+1\right)\,\left(\tan ^2t+1\right)=1$

    $\displaystyle \left(\sin ^2t+1\right)\,\left(\tan ^2t+1\right)=2\,\cos ^2t-1$

  7. Choose the correct trigonometric identity.

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

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

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

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

  8. Choose the correct trigonometric identity.

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

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

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

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

  9. Suppose that $\frac{\pi}{2} < t < \pi$, and that $\displaystyle \csc t=5$ . Find the values of the trigonometric functions $\sin(t)$, $\cos(t)$, and $\tan(t)$.

    $\displaystyle \left[ \sin t={{1}\over{5}} , \cos t={{2\,\sqrt{6}}\over{5}} , \tan t={{1}\over{2\,\sqrt{6}}} \right] $

    $\displaystyle \left[ \sin t={{1}\over{5}} , \cos t=-{{2\,\sqrt{6}}\over{5}} , \tan t=-{{1}\over{2\,\sqrt{6}}} \right] $

    $\displaystyle \left[ \sin t=-{{2\,\sqrt{6}}\over{5}} , \cos t={{1}\over{5}} , \tan t=-2\,\sqrt{6} \right] $

    $\displaystyle \left[ \sin t={{2\,\sqrt{6}}\over{5}} , \cos t={{1}\over{5}} , \tan t=2\,\sqrt{6} \right] $

  10. Find the derivative $f'(x)$ for $f(x) =\displaystyle 2\,\tan x+\sec x $ .

    $f'(x) =\displaystyle -\sec x\,\tan x-2\,\left(\sec x\right)^2 $ $f'(x) =\displaystyle \csc x+2\,\cot x $ $f'(x) =\displaystyle \sec x\,\tan x+2\,\left(\sec x\right)^2 $ $f'(x) =\displaystyle 2\,\left(\csc x\right)^2+\cot x\,\csc x $ $f'(x) =\displaystyle -2\,\left(\csc x\right)^2-\cot x\,\csc x $


Department of Mathematics
Last modified: 2026-10-30