Generating...                               s20quiz10_n6

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

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

  2. Choose the correct trigonometric identity.

    $\sin( t+{{\pi}\over{2}}) = \cos t$

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

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

    $\sin( t+{{\pi}\over{2}}) = \sin t$

  3. Find the derivative $f'(x)$ for $f(x) =\displaystyle -\sin x-2\,\sec x $ .

    $f'(x) =\displaystyle -2\,\csc x-\cos x $ $f'(x) =\displaystyle 2\,\sec x\,\tan x+\cos x $ $f'(x) =\displaystyle \sin x+2\,\cot x\,\csc x $ $f'(x) =\displaystyle -2\,\sec x\,\tan x-\cos x $ $f'(x) =\displaystyle -\sin x-2\,\cot x\,\csc x $

  4. Suppose that $\frac{\pi}{2} < t < \pi$, and that $\displaystyle \csc t=3$ . Find the values of the trigonometric functions $\sin(t)$, $\cos(t)$, and $\tan(t)$.

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

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

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

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

  5. 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] $

  6. Find the derivative $f'(x)$ for $f(x) =\displaystyle \csc x\,\tan x $ .

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

  7. Choose the correct trigonometric identity.

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

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

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

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

  8. Choose the correct trigonometric identity.

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

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

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

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

  9. Suppose that $0 < t < \frac{\pi}{2}$, 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{15}} , \cos t={{1}\over{4}} , \tan t=-{{4 }\over{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] $

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

  10. Choose the correct trigonometric identity.

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

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

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

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


Department of Mathematics
Last modified: 2025-12-16