# Given the values for sin t and cos t, use reciprocal and quotient identities to find the values of the other trigonometric functions of t. sin t = 3/4 and cos t=sqrt7/4

Given the values for sin t and cos t, use reciprocal and quotient identities to find the values of the other trigonometric functions of t.
$\mathrm{sin}t=\frac{3}{4}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\mathrm{cos}t=\frac{\sqrt{7}}{4}$
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Consider the given trigonometric function sint and cost.
The other trigonometric functions values are find as,
Use the trigonometric formula,
$\mathrm{tan}t=\frac{\mathrm{sin}t}{\mathrm{cos}t}$
$\mathrm{csc}t=\frac{1}{\mathrm{sin}t}$
$\mathrm{sec}t=\frac{1}{\mathrm{cos}t}$
$\mathrm{cot}t=\frac{\mathrm{cos}t}{\mathrm{sin}t}$
Now, substitute the given trigonometric function values in the above formula,
Since $\mathrm{sin}t=\frac{3}{4}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\mathrm{cos}t=\frac{\sqrt{7}}{4}$
Thus, $\mathrm{tan}t=\frac{\mathrm{sin}t}{\mathrm{cos}t}=\frac{\frac{3}{4}}{\frac{\sqrt{7}}{4}}=\frac{3}{\sqrt{7}}$
$\mathrm{csc}t=\frac{1}{\mathrm{sin}t}=\frac{1}{\frac{3}{4}}=\frac{4}{3}$
$\mathrm{sec}t=\frac{1}{\mathrm{cos}t}=\frac{1}{\sqrt{\frac{3}{4}}}=\frac{4}{\sqrt{7}}$
$\mathrm{cot}t=\frac{\mathrm{cos}t}{\mathrm{sin}t}=\frac{\frac{\sqrt{7}}{4}}{\frac{3}{4}}=\frac{\sqrt{7}}{3}$

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