Question

the variable t is a real number and p = (\frac{2\sqrt{30}}{11},\frac{1}{11}) is the point on the unit - circle that corresponds to t. find the exact values of the six trigonometric functions of t.
\sin t=
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall trigonometric - function definitions on unit - circle

For a point $P=(x,y)$ on the unit circle corresponding to an angle $t$, $\sin t = y$, $\cos t=x$, $\tan t=\frac{y}{x}(x
eq0)$, $\csc t=\frac{1}{y}(y
eq0)$, $\sec t=\frac{1}{x}(x
eq0)$, $\cot t=\frac{x}{y}(y
eq0)$. Given $P = (\frac{2\sqrt{30}}{11},\frac{1}{11})$, where $x = \frac{2\sqrt{30}}{11}$ and $y=\frac{1}{11}$.

Step2: Find the value of $\sin t$

By the definition of the sine function on the unit - circle, $\sin t=y$.
So, $\sin t=\frac{1}{11}$.

Step3: Find the value of $\cos t$

By the definition of the cosine function on the unit - circle, $\cos t = x$. So, $\cos t=\frac{2\sqrt{30}}{11}$.

Step4: Find the value of $\tan t$

Using the formula $\tan t=\frac{y}{x}$, we substitute $x = \frac{2\sqrt{30}}{11}$ and $y=\frac{1}{11}$: $\tan t=\frac{\frac{1}{11}}{\frac{2\sqrt{30}}{11}}=\frac{1}{2\sqrt{30}}=\frac{\sqrt{30}}{60}$.

Step5: Find the value of $\csc t$

Using the formula $\csc t=\frac{1}{y}$, we substitute $y = \frac{1}{11}$, so $\csc t = 11$.

Step6: Find the value of $\sec t$

Using the formula $\sec t=\frac{1}{x}$, we substitute $x=\frac{2\sqrt{30}}{11}$, so $\sec t=\frac{11}{2\sqrt{30}}=\frac{11\sqrt{30}}{60}$.

Step7: Find the value of $\cot t$

Using the formula $\cot t=\frac{x}{y}$, we substitute $x=\frac{2\sqrt{30}}{11}$ and $y=\frac{1}{11}$, so $\cot t = 2\sqrt{30}$.

Answer:

$\sin t=\frac{1}{11}$, $\cos t=\frac{2\sqrt{30}}{11}$, $\tan t=\frac{\sqrt{30}}{60}$, $\csc t = 11$, $\sec t=\frac{11\sqrt{30}}{60}$, $\cot t = 2\sqrt{30}$