Trigonometry - Trigonometric Ratios of Standard Angles (0°, 30°, 45°, 60°, 90°)

Trigonometric Ratios for standard angles involve calculating the values of $\sin$, $\cos$, $\tan$, $\csc$, $\sec$, and $\cot$ for specific angles: $0^\circ, 30^\circ, 45^\circ, 60^\circ,$ and $90^\circ$. These values are constant and derived from geometric properties.

The ratios for $45^\circ$ are derived from an isosceles right-angled triangle. Visualize a triangle with two equal sides of length $1$ unit and a $90^\circ$ angle between them; using the Pythagoras theorem, the hypotenuse becomes $\sqrt{2}$ units. From this, we see $\sin 45^\circ = \frac{1}{\sqrt{2}}$ and $\cos 45^\circ = \frac{1}{\sqrt{2}}$.

The ratios for $30^\circ$ and $60^\circ$ are derived from an equilateral triangle with side length $2$ units. Imagine splitting this triangle into two equal halves by drawing an altitude. This creates a right-angled triangle with a base of $1$, a hypotenuse of $2$, and an altitude (perpendicular) of $\sqrt{3}$. For the $30^\circ$ angle at the top, the side opposite is $1$, and for the $60^\circ$ angle at the base, the side opposite is $\sqrt{3}$.

As the angle $\theta$ increases from $0^\circ$ to $90^\circ$, the value of $\sin \theta$ increases from $0$ to $1$, while the value of $\cos \theta$ decreases from $1$ to $0$. This reflects the change in the lengths of the opposite and adjacent sides relative to the hypotenuse in a unit circle.

The Tangent ratio ($\tan \theta$) is the quotient of Sine and Cosine ($\tan \theta = \frac{\sin \theta}{\cos \theta}$). At $90^\circ$, $\cos 90^\circ = 0$, making $\tan 90^\circ$ undefined or 'Not Defined' because division by zero is impossible.

Reciprocal Trigonometric Ratios are defined as: $\csc \theta$ is the reciprocal of $\sin \theta$, $\sec \theta$ is the reciprocal of $\cos \theta$, and $\cot \theta$ is the reciprocal of $\tan \theta$. For example, since $\sin 30^\circ = \frac{1}{2}$, its reciprocal $\csc 30^\circ = 2$.