Introduction to Trigonometry - Trigonometric Ratios of Specific Angles (0°, 30°, 45°, 60°, 90°)

Trigonometric ratios are defined based on the relationship between the angles and the sides of a right-angled triangle. For specific angles like $30^\circ, 45^\circ$, and $60^\circ$, these ratios are derived from geometric constructions such as equilateral and isosceles right triangles.

For $45^\circ$, we consider an isosceles right triangle where the base and perpendicular are equal ($a$). By Pythagoras theorem, the hypotenuse is $a\sqrt{2}$. This visualizes why $\sin 45^\circ$ and $\cos 45^\circ$ are both equal to $\frac{1}{\sqrt{2}}$.

For $30^\circ$ and $60^\circ$, we visualize an equilateral triangle with side $2a$. When an altitude is drawn, it bisects the base and the vertex angle, creating a $30^\circ-60^\circ-90^\circ$ triangle with sides $a$, $a\sqrt{3}$, and $2a$. This explains why $\sin 30^\circ = \frac{1}{2}$ and $\cos 30^\circ = \frac{\sqrt{3}}{2}$.

At $0^\circ$, we imagine the angle at the base of a right triangle shrinking. As the angle approaches $0^\circ$, the length of the perpendicular (opposite side) becomes $0$, and the hypotenuse overlaps the base. Thus, $\sin 0^\circ = 0$ and $\cos 0^\circ = 1$.

At $90^\circ$, we imagine the angle at the base increasing. As it approaches $90^\circ$, the base (adjacent side) becomes $0$, and the hypotenuse overlaps the perpendicular. Thus, $\sin 90^\circ = 1$ and $\cos 90^\circ = 0$.

The $\tan \theta$ ratio for these specific angles is always the ratio of $\sin \theta$ to $\cos \theta$. For instance, at $90^\circ$, since $\cos 90^\circ = 0$, the value of $\tan 90^\circ$ is undefined ($1/0$).

The reciprocal ratios ($\text{cosec}$, $\sec$, $\cot$) are derived by inverting the values of $\sin$, $\cos$, and $\tan$ respectively. If a ratio is $0$, its reciprocal is undefined; if a ratio is undefined, its reciprocal is $0$.