Generating...                               s20quiz10_n26

  1. Find the derivative $f'(x)$ for $f(x) =\displaystyle {{\sin x}\over{x}} $ .

    $f'(x) =\displaystyle {{x\,\sin x+\cos x}\over{x^2}} $ $f'(x) =\displaystyle {{\sin x-x\,\cos x}\over{x^2}} $ $f'(x) =\displaystyle -{{x\,\sin x+\cos x}\over{x^2}} $ $f'(x) =\displaystyle -{{\sin x-x\,\cos x}\over{x^2}} $

  2. Choose the correct trigonometric identity.

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

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

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

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

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

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

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

  4. Choose the correct trigonometric identity.

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

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

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

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

  5. Choose the correct trigonometric identity.

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

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

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

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

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

    $f'(x) =\displaystyle \cot x\,\csc x-3\,\left(\sec x\right)^2 $ $f'(x) =\displaystyle -3\,\tan x-\csc x $ $f'(x) =\displaystyle 3\,\left(\csc x\right)^2-\sec x\,\tan x $ $f'(x) =\displaystyle 3\,\left(\sec x\right)^2-\cot x\,\csc x $ $f'(x) =\displaystyle \sec x\,\tan x-3\,\left(\csc x\right)^2 $

  7. Find the values of $\displaystyle \sin t$ and $\displaystyle \cos t$ for $\displaystyle t={{4\,\pi}\over{3}}$ .

    $\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] $

  8. Choose the correct trigonometric identity.

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

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

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

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

  9. Choose the correct trigonometric identity.

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

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

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

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

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

    $\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={{\sqrt{3}}\over{2}} , \cos t=-{{1}\over{2}} , \tan t= -\sqrt{3} \right] $

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


Department of Mathematics
Last modified: 2026-03-24