Question

the variable t is a real number and p = (\frac{\sqrt{3}}{5},\frac{\sqrt{22}}{5}) is the point on the unit - circle that corresponds to t. find the exact values of the six trigonometric functions of t.
\sin t = \frac{\sqrt{22}}{5}
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
\cos t=
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall trig - function definitions on unit circle

For a point $P(x,y)$ on the unit circle corresponding to a real - number $t$, $\sin t=y$, $\cos t = x$, $\tan t=\frac{y}{x}(x
eq0)$, $\csc t=\frac{1}{y}(y
eq0)$, $\sec t=\frac{1}{x}(x
eq0)$, $\cot t=\frac{x}{y}(y
eq0)$. Given $P = (\frac{\sqrt{3}}{5},\frac{\sqrt{22}}{5})$, where $x = \frac{\sqrt{3}}{5}$ and $y=\frac{\sqrt{22}}{5}$.

Step2: Find the value of $\cos t$

Since $\cos t=x$, then $\cos t=\frac{\sqrt{3}}{5}$.

Step3: Find the value of $\tan t$

$\tan t=\frac{y}{x}=\frac{\frac{\sqrt{22}}{5}}{\frac{\sqrt{3}}{5}}=\frac{\sqrt{22}}{\sqrt{3}}=\frac{\sqrt{66}}{3}$.

Step4: Find the value of $\csc t$

$\csc t=\frac{1}{y}=\frac{5}{\sqrt{22}}=\frac{5\sqrt{22}}{22}$.

Step5: Find the value of $\sec t$

$\sec t=\frac{1}{x}=\frac{5}{\sqrt{3}}=\frac{5\sqrt{3}}{3}$.

Step6: Find the value of $\cot t$

$\cot t=\frac{x}{y}=\frac{\frac{\sqrt{3}}{5}}{\frac{\sqrt{22}}{5}}=\frac{\sqrt{3}}{\sqrt{22}}=\frac{\sqrt{66}}{22}$.

Answer:

$\cos t=\frac{\sqrt{3}}{5}$, $\tan t=\frac{\sqrt{66}}{3}$, $\csc t=\frac{5\sqrt{22}}{22}$, $\sec t=\frac{5\sqrt{3}}{3}$, $\cot t=\frac{\sqrt{66}}{22}$