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.)
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Solution

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Solution Steps

Step 1: Identify the Coordinates

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

Step 2: Calculate ant an t

Using the definition ant=sintcost=0.50.866=0.577 an t = \frac{\sin t}{\cos t} = \frac{-0.5}{0.866} = -0.577.

Step 3: Calculate csct,sect,cott\csc t, \sec t, \cot t

Using the reciprocal identities, csct=1sint=10.5=2\csc t = \frac{1}{\sin t} = \frac{1}{-0.5} = -2 and cott=1ant=costsint=0.8660.5=1.732\cot t = \frac{1}{ an t} = \frac{\cos t}{\sin t} = \frac{0.866}{-0.5} = -1.732.

Also, sect=1cost=10.866=1.155\sec t = \frac{1}{\cos t} = \frac{1}{0.866} = 1.155.

Final Answer:

sint=0.5,cost=0.866,ant=0.577,csct=2,sect=1.155,cott=1.732\sin t = -0.5, \cos t = 0.866, an t = -0.577, \csc t = -2, \sec t = 1.155, \cot t = -1.732

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