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

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

    $f'(x) =\displaystyle -3\,\sin x-2\,\sec x $ $f'(x) =\displaystyle 3\,\sin x+2\,\cot x\,\csc x $ $f'(x) =\displaystyle -3\,\sin x-2\,\cot x\,\csc x $ $f'(x) =\displaystyle 2\,\sec x\,\tan x+3\,\cos x $ $f'(x) =\displaystyle -2\,\sec x\,\tan x-3\,\cos x $

  3. Suppose that $\displaystyle g\left( {{5\,\pi}\over{3}} \right) = -1 $ and $\displaystyle g'\left( {{5\,\pi}\over{3}} \right) = 2 $. Then find the derivative $\displaystyle f'\left( {{5\,\pi}\over{3}} \right)$ for $f(x) = \displaystyle g\left(x\right)\,\tan x $ .

    $\displaystyle -2\,\sqrt{3}-4$ $\displaystyle \sqrt{3}$ $\displaystyle 4-2\,\sqrt{3}$ 2 $\displaystyle 2\,\sqrt{3}-4$

  4. Find the derivative $f'(x)$ for $f(x) =\displaystyle {{\cot x}\over{x^2}} $ .

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

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

    $\displaystyle \left[ \sin t={{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=-{{1}\over{8}} , \tan t={{8 }\over{3}} \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=-{{1}\over{8}} , \cos t=-{{1}\over{3}} , \tan t={{3 }\over{8}} \right] $

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

    $f'(x) =\displaystyle e^{x}\,\left(\sin x-\cos x\right) $ $f'(x) =\displaystyle -e^{x}\,\left(\sin x+\cos x\right) $ $f'(x) =\displaystyle e^{x}\,\left(\sin x+\cos x\right) $ $f'(x) =\displaystyle -e^{x}\,\left(\sin x-\cos x\right) $

  7. Choose the correct trigonometric identity.

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

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

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

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

  8. 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] $

  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={{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] $

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

  10. Choose the correct trigonometric identity.

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

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

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

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


Department of Mathematics
Last modified: 2026-07-06