Given that ( heta ) is an angle in the unit circle such that its terminal side passes through the point ((a,b)). If the line passing through ((a,b)) and the origin makes an angle ( alpha ) with the x-axis, find the values of ( sin(alpha) ),

Given that \( \theta \) is an angle in the unit circle such that its terminal side passes through the point \((a,b)\):

The coordinates \((a, b)\) on the unit circle imply that \(a = \cos(\theta)\) and \(b = \sin(\theta)\).

Since the line passing through \((a, b)\) and the origin makes an angle \( \alpha \) with the x-axis:

$ \sin(\alpha) = \frac{b}{\sqrt{a^2 + b^2}} $

$ \cos(\alpha) = \frac{a}{\sqrt{a^2 + b^2}} $

$ \tan(\alpha) = \frac{b}{a} $

Given that \( \theta \) is in the second quadrant:

$ \theta = \pi – \alpha $

Given that ( heta ) is an angle in the unit circle such that its terminal side passes through the point ((a,b)):

The coordinates ((a, b)) on the unit circle imply that (a = cos( heta)) and (b = sin( heta)).

Since the line passing through ((a, b)) and the origin makes an angle ( alpha ) with the x-axis:

$ sin(alpha) = b ext{ (since the unit circle radius is 1)} $

$ cos(alpha) = sqrt{1 – b^2} $

$ an(alpha) = frac{b}{sqrt{1 – b^2}} $

Given that ( heta ) is in the second quadrant:

$ heta = pi – alpha $

Given that ( heta ) is an angle in the unit circle such that its terminal side passes through the point ((a,b)):

$ sin(alpha) = frac{b}{sqrt{a^2 + b^2}} $

$ cos(alpha) = frac{a}{sqrt{a^2 + b^2}} $

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

In the second quadrant:

$ heta = pi – alpha $

Start Using PopAi Today