Find the values of $ sin(A) $, $ cos(B) $, and $ an(C) $ on the unit circle given specific conditions

Consider the unit circle centered at the origin $(0,0)$ in the coordinate plane. Given that $A$, $B$, and $C$ are angles in the unit circle, find $\sin(A)$, $\cos(B)$, and $\tan(C)$ if the following conditions are met:

1) $A = \pi/3$

2) $B = 3\pi/4$

3) $C = 5\pi/6$

Answer:

1) For $A = \pi/3$:

$ \sin(A) = \sin(\pi/3) = \frac{\sqrt{3}}{2} $

2) For $B = 3\pi/4$:

$ \cos(B) = \cos(3\pi/4) = -\frac{\sqrt{2}}{2} $

3) For $C = 5\pi/6$:

$ \tan(C) = \tan(5\pi/6) = -\frac{1}{\sqrt{3}} $

Consider the unit circle with angles $ heta_1$, $ heta_2$, and $ heta_3$ such that:

1) $ heta_1 = pi/6$

2) $ heta_2 = pi/4$

3) $ heta_3 = 2pi/3$

Find the values of $sin( heta_1)$, $cos( heta_2)$, and $ an( heta_3)$. Answer:

1) For $ heta_1 = pi/6$:

$ sin( heta_1) = sin(pi/6) = frac{1}{2} $

2) For $ heta_2 = pi/4$:

$ cos( heta_2) = cos(pi/4) = frac{sqrt{2}}{2} $

3) For $ heta_3 = 2pi/3$:

$ an( heta_3) = an(2pi/3) = -sqrt{3} $

Given the angles $alpha = pi/2$, $eta = pi/4$, and $gamma = 7pi/6$ on the unit circle, find:

1) $sin(alpha)$:

$ sin(alpha) = sin(pi/2) = 1 $

2) $cos(eta)$:

$ cos(eta) = cos(pi/4) = frac{sqrt{2}}{2} $

3) $ an(gamma)$:

$ an(gamma) = an(7pi/6) = frac{1}{sqrt{3}} $

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