Determine the $sin$ value from the unit circle and verify identities

Given the point $P(-\frac{1}{2}, -\frac{\sqrt{3}}{2})$ on the unit circle, determine $\sin(\theta)$ and verify the identity $\sin^2(\theta) + \cos^2(\theta) = 1$:

1. Identify the coordinates of point $P$ as $(\cos(\theta), \sin(\theta))$.

2. From $P(-\frac{1}{2}, -\frac{\sqrt{3}}{2})$, we have $\cos(\theta) = -\frac{1}{2}$ and $\sin(\theta) = -\frac{\sqrt{3}}{2}$.

3. Verify the identity:

$\sin^2(\theta) + \cos^2(\theta) = \left(-\frac{\sqrt{3}}{2}\right)^2 + \left(-\frac{1}{2}\right)^2$

$= \frac{3}{4} + \frac{1}{4} = 1$

The identity is verified as true.

Given point $P(-frac{1}{2}, -frac{sqrt{3}}{2})$ on the unit circle, calculate $sin( heta)$ and confirm the identity $sin^2( heta) + cos^2( heta) = 1$:

1. Point $P$ is identified as $(cos( heta), sin( heta))$.

2. Hence, $cos( heta) = -frac{1}{2}$ and $sin( heta) = -frac{sqrt{3}}{2}$.

3. Verification:

$sin^2( heta) + cos^2( heta) = left(-frac{sqrt{3}}{2}
ight)^2 + left(-frac{1}{2}
ight)^2$

$= frac{3}{4} + frac{1}{4} = 1$

The identity holds true.

From point $P(-frac{1}{2}, -frac{sqrt{3}}{2})$ on the unit circle, find $sin( heta)$ and check $sin^2( heta) + cos^2( heta) = 1$:

1. $cos( heta) = -frac{1}{2}$ and $sin( heta) = -frac{sqrt{3}}{2}$.

2. Verify:

$sin^2( heta) + cos^2( heta) = left(-frac{sqrt{3}}{2}
ight)^2 + left(-frac{1}{2}
ight)^2$

$= frac{3}{4} + frac{1}{4} = 1$

The identity is confirmed as true.

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