Trigonometric FunctionsClass 11 Mathematics Important Points

A full circle is $360^{\circ}$ or $2\pi$ radians. The key conversion relationship is $\pi \text{ radians} = 180^{\circ}$.

To convert from degrees to radians, multiply by $\frac{\pi}{180}$. To convert from radians to degrees, multiply by $\frac{180}{\pi}$.

For a circle of radius $r$, an arc of length $l$ subtends a central angle $\theta$ (in radians) given by the formula $l = r\theta$.

For a point $P(a, b)$ on the unit circle corresponding to an angle $x$, the definitions are $\cos x = a$ and $\sin x = b$. This leads to the fundamental Pythagorean identity $\cos^2 x + \sin^2 x = 1$.

In Quadrant I, all are positive. In II, only $\sin x$ and $\csc x$ are positive. In III, only $\tan x$ and $\cot x$ are positive. In IV, only $\cos x$ and $\sec x$ are positive.

The domain for $\sin x$ and $\cos x$ is all real numbers $\mathbf{R}$ and the range is $[-1, 1]$. The range of $\tan x$ is $\mathbf{R}$.

The sine and cosine functions are periodic with a period of $2\pi$, so $\sin(x + 2\pi) = \sin x$. The tangent function is periodic with a period of $\pi$, so $\tan(x + \pi) = \tan x$.

Cosine is an even function, $\cos(-x) = \cos x$. Sine and tangent are odd functions, $\sin(-x) = -\sin x$ and $\tan(-x) = -\tan x$.

The identities for cosine are $\cos(x+y) = \cos x \cos y - \sin x \sin y$ and $\cos(x-y) = \cos x \cos y + \sin x \sin y$.

The identities for sine are $\sin(x+y) = \sin x \cos y + \cos x \sin y$ and $\sin(x-y) = \sin x \cos y - \cos x \sin y$.

The identity for tangent of a sum is $\tan(x+y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}$. For a difference, it is $\tan(x-y) = \frac{\tan x - \tan y}{1 + \tan x \tan y}$.

Key double angle formulas are $\sin 2x = 2\sin x \cos x$ and $\cos 2x = \cos^2 x - \sin^2 x$. The tangent formula is $\tan 2x = \frac{2\tan x}{1 - \tan^2 x}$.

The $\cos 2x$ formula can also be written as $\cos 2x = 2\cos^2 x - 1$ or $\cos 2x = 1 - 2\sin^2 x$.

The formulas for triple angles are $\sin 3x = 3\sin x - 4\sin^3 x$ and $\cos 3x = 4\cos^3 x - 3\cos x$.

These convert sums to products, for example: $\sin x + \sin y = 2\sin(\frac{x+y}{2})\cos(\frac{x-y}{2})$ and $\cos x + \cos y = 2\cos(\frac{x+y}{2})\cos(\frac{x-y}{2})$.

These convert products to sums, for example: $2\cos x \cos y = \cos(x+y) + \cos(x-y)$ and $2\sin x \cos y = \sin(x+y) + \sin(x-y)$.