For an angle \(\theta\) measured from the positive x axis counter-clockwise on the unit circle (radius = 1), the coordinates of the intersection point with the circumference are \((\cos\theta,\;\sin\theta)\).
1. Sine \(\bigl(\sin\theta\bigr)\)
- Geometric definition: projection of the radius onto the y-axis. In a right triangle, \(\sin\theta=\dfrac{\text{opposite leg}}{\text{hypotenuse}}\).
- Domain: \(\mathbb R\). Range: \([-1,1]\).
- Period: \(2\pi\). Parity: odd function \(\bigl(\sin(-\theta)=-\sin\theta\bigr)\).
2. Cosine \(\bigl(\cos\theta\bigr)\)
- Geometric definition: projection of the radius onto the x-axis. In a right triangle, \(\cos\theta=\dfrac{\text{adjacent leg}}{\text{hypotenuse}}\).
- Domain: \(\mathbb R\). Range: \([-1,1]\).
- Period: \(2\pi\). Parity: even function \(\bigl(\cos(-\theta)=\cos\theta\bigr)\).
3. Tangent \(\bigl(\tan\theta\bigr)\)
- Definition: ratio between sine and cosine: \(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\). Geometrically, it is the slope of the line through the origin and the point \((1,\tan\theta)\) where the extended radius intersects the vertical tangent to the circle at \(x=1\).
- Domain: \(\theta\ne\dfrac{\pi}{2}+k\pi\) (where \(\cos\theta=0\)). Range: \(\mathbb R\).
- Period: \(\pi\). Parity: odd function.
Fundamental relationships
Pythagorean identity: \(\sin^{2}\theta+\cos^{2}\theta=1\).
Tangent in terms of sine and cosine: \(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\).
Complementary angles: \(\sin\theta=\cos\left(\dfrac{\pi}{2}-\theta\right)\).
Note: All properties generalise to any radius by multiplying the projections by \(R\); on the unit circle \(R=1\), which simplifies the expressions.