a

cos

- Sine and cosine are periodic functions of period $360^{\circ}$, that is, of period $2\pi $.
That's because sines and cosines are defined in terms of angles, and you can add
multiples of $360^{\circ}$, or $2\pi $, and it doesn't change the angle. Thus, for any angle x

$sin(x+2 \pi)= sin (x)$ and $cos(x+2 \pi)= cos(x)$

or we can say that

$sin (2n \pi + x) = sin x$, $n\in Z$ , $cos (2n \pi + x) = cos x$, $n\in Z$

Where Z is the set of all integers - sin x = 0 implies $x = n \pi $ , where n is any integer

cos x = 0 implies $x = (2n + 1)\frac {\pi}{2}$ - The other trigonometric function are defined as

$cosec(x)= \frac {1}{sin (x)}$ where $x \neq n \pi$, where n is any integer

$sec(x)=\frac {1}{cos (x)}$ where $x \neq (2n + 1)\frac {\pi}{2}$ where n is any integer

$tan(x)=\frac {sin(x)}{cos(x)} $ where $x \neq (2n + 1)\frac {\pi}{2}$ where n is any integer

$cot(x)=\frac {cos(x)}{sin(x)}$ where $x \neq n \pi$, where n is any integer -
**Trigonometric Functions identities**

For all real x

$sin ^2(x)+cos ^2(x)=1$

$1+ tan ^2(x)=sec ^2(x)$

$1+ cot ^2(x)=cosec ^2(x)$ **What is is Odd function and Even Function**

We have come across these adjectives 'odd' and 'even' when applied to functions, but it's important to know them. A function f is said to be an odd function

if for any number x,*f(-x) = -f(x)*.

A function f is said to be an even function if for any number x,*f(-x) = f(x)*.

Many functions are neither odd nor even functions, but some of the most important functions are one or the other.

Example:

Any polynomial with only odd degree terms is an odd function, for example, f(x) = 2x^{7}+ 9x^{5}- x. (Note that all the powers of x are odd numbers.)

Similarly, any polynomial with only even degree terms is an even function. For example, f(x) = 6x^{8}- 6x^{2}- 5.

Based on above definition we can call Sine is an odd function, and cosine is even

sin (-x) = -sin x, and

cos (-x) = cos x.

These facts follow from the symmetry of the unit circle across the x-axis. The angle -x is the same angle as x except it's on the other side of the x-axis. Flipping a point (x,y) to the other side of the x-axis makes it into (x,-y), so the y-coordinate is negated, that is, the sine is negated, but the x-coordinate remains the same, that is, the cosine is unchanged.**Sign of the Trigonometric functions**Now since in unit circle

$-1 \leq a \leq 1$

$-1 \leq b \leq 1$

It follows that for all x

$-1 \leq sin(x) \leq 1$

$-1 \leq cos(x) \leq 1$

Also We know from previous classes,

(i) a,b are both positive in Ist quadrant i.e $ 0< x < \frac {\pi}{2}$ , It implies that sin is positive and cos is positive

(ii) a is negative and b is positive in IInd quadrant i.e $\frac {\pi}{2} < x < \pi $ ,It implies that sin is negative and cos is positive

(iii) a and b both are negative in III quadrant ie. $ \pi < x < \frac {3 \pi}{2} $ ,It implies that sin is negative and cos is negative

(iv) a is positive and b is negative in IV quadrant i,.e $\frac {3 \pi}{2} < x < 2 \pi $ It implies that sin is positive and cos is negative

Similarly sign can be obtained for other functions

This can be summarized in table as below

This can be remember as**ALL SILVER TEA CUPS**

**A**LL -> ALL are positive in Ist quadrant

**S**ILVER -> Sin function and its reciprocal function are positive

**T**EA -> Tangent function and its reciprocak function are positive

**C**UPS -> Cos function and its reciprocal function are positive

- The common values are same as trigonometric ratios for those angles

- The relation for complementary angles and ssupplementary angles are given below

Class 11 Maths Class 11 Physics