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multiple choice testing csc, sec, cot  of  random(pi/6, pi/4, pi/3) radians

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Often we prefer to work with exact values rather than approximations from a calculator.

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Recall that $\\csc\\theta=\\dfrac{1}{\\sin\\theta}$, $\\sec\\theta=\\dfrac{1}{\\cos\\theta}$, and $\\cot\\theta=\\dfrac{1}{\\tan\\theta}$.

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By drawing the following triangles we can determine the exact values of $\\sin$, $\\cos$ and $\\tan$ (and their reciprocals $\\csc$, $\\sec$, $\\cot$) for the angles $\\dfrac{\\pi}{6}$, $\\dfrac{\\pi}{4}$ and $\\dfrac{\\pi}{3}$.

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$\\csc\\left(\\var{distheta}\\right)=\\dfrac{\\text{Hypotenuse}}{\\text{Opposite}}=\\;\\;2$$\\csc\\left(\\var{distheta}\\right)=\\dfrac{\\text{Hypotenuse}}{\\text{Opposite}}=\\sqrt{2}$$\\csc\\left(\\var{distheta}\\right)=\\dfrac{\\text{Hypotenuse}}{\\text{Opposite}}=\\dfrac{2}{\\sqrt{3}}$

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$\\sec\\left(\\var{distheta}\\right)=\\dfrac{\\text{Hypotenuse}}{\\text{Adjacent}}=\\dfrac{2}{\\sqrt{3}}$$\\sec\\left(\\var{distheta}\\right)=\\dfrac{\\text{Hypotenuse}}{\\text{Adjacent}}=\\sqrt{2}$$\\sec\\left(\\var{distheta}\\right)=\\dfrac{\\text{Hypotenuse}}{\\text{Adjacent}}=\\;\\;2$

\n

$\\cot\\left(\\var{distheta}\\right)=\\;\\;\\dfrac{\\text{Adjacent}}{\\text{Opposite}}\\;\\;=\\sqrt{3}$$\\cot\\left(\\var{distheta}\\right)=\\;\\;\\dfrac{\\text{Adjacent}}{\\text{Opposite}}\\;\\;=\\;\\;1$$\\cot\\left(\\var{distheta}\\right)=\\;\\;\\dfrac{\\text{Adjacent}}{\\text{Opposite}}\\;\\;=\\dfrac{1}{\\sqrt{3}}$

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Alternatively, one can memorise the following table: 

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\dfrac{\\pi}{6}$$\\dfrac{\\pi}{4}$$\\dfrac{\\pi}{3}$
 
$\\sin$$\\dfrac{1}{2}$$\\dfrac{1}{\\sqrt{2}}$$\\dfrac{\\sqrt{3}}{2}$
 
$\\cos$$\\dfrac{\\sqrt{3}}{2}$$\\dfrac{1}{\\sqrt{2}}$$\\dfrac{1}{2}$
 
$\\tan$$\\dfrac{1}{\\sqrt{3}}$$1$$\\sqrt{3}$
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Since we are asked about $\\var{distheta}$, we use the $\\var{distheta}$ column of the table to determine that:

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$\\csc\\left(\\var{distheta}\\right)=\\dfrac{1}{\\sin\\left(\\var{distheta}\\right)}=\\;\\;2$$\\csc\\left(\\var{distheta}\\right)=\\dfrac{1}{\\sin\\left(\\var{distheta}\\right)}=\\sqrt{2}$$\\csc\\left(\\var{distheta}\\right)=\\dfrac{1}{\\sin\\left(\\var{distheta}\\right)}=\\dfrac{2}{\\sqrt{3}}$

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$\\sec\\left(\\var{distheta}\\right)=\\dfrac{1}{\\cos\\left(\\var{distheta}\\right)}=\\dfrac{2}{\\sqrt{3}}$$\\sec\\left(\\var{distheta}\\right)=\\dfrac{1}{\\cos\\left(\\var{distheta}\\right)}=\\sqrt{2}$$\\sec\\left(\\var{distheta}\\right)=\\dfrac{1}{\\cos\\left(\\var{distheta}\\right)}=\\;\\;2$

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$\\cot\\left(\\var{distheta}\\right)=\\dfrac{1}{\\tan\\left(\\var{distheta}\\right)}=\\sqrt{3}$$\\cot\\left(\\var{distheta}\\right)=\\dfrac{1}{\\tan\\left(\\var{distheta}\\right)}=\\;\\;1$$\\cot\\left(\\var{distheta}\\right)=\\dfrac{1}{\\tan\\left(\\var{distheta}\\right)}=\\dfrac{1}{\\sqrt{3}}$

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The exact value of $\\csc\\Large($$\\var{distheta}\\Large)$ is

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$\\dfrac{1}{\\sqrt{3}}$

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$1$

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The exact value of $\\sec\\Large($$\\var{distheta}\\Large)$ is

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$\\sqrt{3}$

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$\\dfrac{1}{\\sqrt{3}}$

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$1$

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The exact value of $\\cot\\Large($$\\var{distheta}\\Large)$ is

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$\\sqrt{3}$

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$\\dfrac{1}{\\sqrt{3}}$

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$1$

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