Given the point $P(a, b)$ on the unit circle, find the exact values of sine, cosine, and tangent for the angles $ heta$ and $phi$ where $ heta$ is the angle between the positive x-axis and the line segment $OP$ and $phi$ is the angle in radians corres

Given the point $P(a, b)$ on the unit circle, we know that $a^2 + b^2 = 1$.

For angle $\theta$:

The sine and cosine values are the coordinates of point P, so:

$\sin(\theta) = b$

$\cos(\theta) = a$

To find the tangent, we use:

$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{b}{a}$

For angle $\phi$:

Since $\phi$ represents the arc length from $(1, 0)$ to $P$, we use the unit circle property that $\phi$ forms the same angle as $\theta$ from the origin:

$\sin(\phi) = b$

$\cos(\phi) = a$

$\tan(\phi) = \frac{\sin(\phi)}{\cos(\phi)} = \frac{b}{a}$

To solve for $ heta$ and $phi$ given $P(a, b)$ on the unit circle, we start with:

$a^2 + b^2 = 1$

For $ heta$:

$sin( heta) = b$

$cos( heta) = a$

$ an( heta) = frac{b}{a}$

For $phi$:

Since $phi$ is the arc length corresponding to the angle $ heta$:

$phi = heta$

So,

$sin(phi) = b$

$cos(phi) = a$

$ an(phi) = frac{b}{a}$

Given $P(a, b)$ on the unit circle,

$a^2 + b^2 = 1$

For $ heta$:

$sin( heta) = b$

$cos( heta) = a$

$ an( heta) = frac{b}{a}$

For $phi$:

$sin(phi) = b$

$cos(phi) = a$

$ an(phi) = frac{b}{a}$

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