Questions: Find the values of sin t, cos t, tan t, csc t, sec t, and cot t if P=(sqrt(3)/2, -1/2) 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.)

$Find the values of sin t, cos t, tan t, csc t, sec t, and cot t if P=(sqrt(3)/2, -1/2) is the point on the unit circle that corresponds to the real number t. sin t= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)$

Transcript text: Find the values of $\sin t, \cos t, \tan t, \csc t, \sec t$, and $\cot t$ if $P=\left(\frac{\sqrt{3}}{2},-\frac{1}{2}\right)$ 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.)

Solution

Solution Steps

Step 1: Identify the Coordinates

On the unit circle, the coordinates of point $P=(0.866,-0.5)$ directly give us $\cos t = 0.866$ and $\sin t = -0.5$.

Step 2: Calculate $ an t$

Using the definition $ an t = \frac{\sin t}{\cos t} = \frac{-0.5}{0.866} = -0.577$.

Step 3: Calculate $\csc t, \sec t, \cot t$

Using the reciprocal identities, $\csc t = \frac{1}{\sin t} = \frac{1}{-0.5} = -2$ and $\cot t = \frac{1}{ an t} = \frac{\cos t}{\sin t} = \frac{0.866}{-0.5} = -1.732$.

Also, $\sec t = \frac{1}{\cos t} = \frac{1}{0.866} = 1.155$.