Question

find the values of sin t, cos t, tan t, csc t, sec t, and cot t if p = (1/2, √3/2) is the point on the unit circle that corresponds to the real number t. sin 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 real - number $t$, $\sin t=y$, $\cos t = x$, $\tan t=\frac{y}{x}(x
eq0)$, $\csc t=\frac{1}{y}(y
eq0)$, $\sec t=\frac{1}{x}(x
eq0)$, and $\cot t=\frac{x}{y}(y
eq0)$.
Given $P = (\frac{1}{2},\frac{\sqrt{3}}{2})$, where $x = \frac{1}{2}$ and $y=\frac{\sqrt{3}}{2}$.

Step2: Calculate $\sin t$

By the definition $\sin t=y$.
So, $\sin t=\frac{\sqrt{3}}{2}$.

Step3: Calculate $\cos t$

By the definition $\cos t=x$.
So, $\cos t=\frac{1}{2}$.

Step4: Calculate $\tan t$

$\tan t=\frac{y}{x}=\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\sqrt{3}$.

Step5: Calculate $\csc t$

$\csc t=\frac{1}{y}=\frac{1}{\frac{\sqrt{3}}{2}}=\frac{2\sqrt{3}}{3}$.

Step6: Calculate $\sec t$

$\sec t=\frac{1}{x}=\frac{1}{\frac{1}{2}} = 2$.

Step7: Calculate $\cot t$

$\cot t=\frac{x}{y}=\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}=\frac{\sqrt{3}}{3}$.

Answer:

$\sin t=\frac{\sqrt{3}}{2}$, $\cos t=\frac{1}{2}$, $\tan t=\sqrt{3}$, $\csc t=\frac{2\sqrt{3}}{3}$, $\sec t = 2$, $\cot t=\frac{\sqrt{3}}{3}$