Find the coordinates of a point and the corresponding angle on the unit circle given the sine value, and prove if the cosine value meets the trigonometric identity.

Given the sine value $\sin(\theta) = \frac{3}{5}$ on the unit circle, find the coordinates $(x,y)$ of the point and the angle $\theta$. Verify if the cosine value $\cos(\theta)$ satisfies the trigonometric identity.

Step 1: Use the Pythagorean identity:

$\sin^2(\theta) + \cos^2(\theta) = 1$

Step 2: Substitute $\sin(\theta) = \frac{3}{5}$ into the identity:

$\left(\frac{3}{5}\right)^2 + \cos^2(\theta) = 1$

$\frac{9}{25} + \cos^2(\theta) = 1$

Step 3: Solve for $\cos^2(\theta)$:

$\cos^2(\theta) = 1 – \frac{9}{25}$

$\cos^2(\theta) = \frac{16}{25}$

Step 4: Determine $\cos(\theta)$:

$\cos(\theta) = \pm \frac{4}{5}$

Step 5: Verify the coordinates:

The coordinates are $(\pm \frac{4}{5}, \frac{3}{5})$ for $\theta = \arcsin(\frac{3}{5})$.

Given $sin( heta) = frac{3}{5}$, find $(x,y)$ and angle $ heta$. Verify $cos( heta)$ satisfies the identity.

1. Pythagorean identity:

$sin^2( heta) + cos^2( heta) = 1$

2. Substitute $sin( heta) = frac{3}{5}$:

$left(frac{3}{5}
ight)^2 + cos^2( heta) = 1$

$frac{9}{25} + cos^2( heta) = 1$

3. Solve for $cos^2( heta)$:

$cos^2( heta) = 1 – frac{9}{25}$

$cos^2( heta) = frac{16}{25}$

4. Determine $cos( heta)$:

$cos( heta) = pm frac{4}{5}$

The coordinates are $(pm frac{4}{5}, frac{3}{5})$ for $ heta = arcsin(frac{3}{5})$.

Given $sin( heta) = frac{3}{5}$, find $(x,y)$ and angle $ heta$. Verify $cos( heta)$.

1. $sin^2( heta) + cos^2( heta) = 1$

2. $left(frac{3}{5}
ight)^2 + cos^2( heta) = 1$

3. Solve for $cos( heta)$:

$cos^2( heta) = frac{16}{25}$

$cos( heta) = pm frac{4}{5}$

The coordinates are $(pm frac{4}{5}, frac{3}{5})$.

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