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+2π)= sin (x) and cos(x+2π)= cos(x)

or we can say that

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

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

cos x = 0 implies x = (2n + 1)(π/2) - The other trignometric function are defined as

cosec(x)= 1/sin (x) where x ≠nπ, where n is any integer

sec(x)=1/cos (x) where x ≠(2n + 1)(π/2) where n is any integer

tan(x)=sin(x)/cos(x) where x ≠(2n + 1)(π/2) where n is any integer

cot(x)=cos(x)/sin(x) where x ≠nπ, where n is any integer - 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 defination 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. - Now since in unit circle

-1 ≤ a ≤ 1

-1 ≤ b ≤ 1

It follows that for all x

-1 ≤ sin(x) ≤ 1

-1 ≤ cos(x) ≤ 1 Also We know from previous classes,

a,b are both positive in Ist quadrant i.e 0< x < π/2 It implies that sin is positive and cos is postive

a is negative and b is positive in IInd quadrant i.e π/2 < x< πIt implies that sin is negative and cos is postive

a and b both are negative in III quadrant ie. π < x < 3π/2 It implies that sin is negative and cos is negative

a is positive and b is negative in IV quadrant i,.e 3π/2 < x < 2π It implies that sin is positive and cos is negative

Similarly sign can be obtained for other functions

Go Back to Class 11 Maths Home page Go Back to Class 11 Physics Home page

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