Trigonometric Functions

Trigonometric Ratios

Let a rotating ray starts from initial position OA and move about O and trace out an angle ∠AOB = x. Consider the right angled triangle OAB.

sin x = $\large \frac{opposite}{hypotenuse}$

cos x = $\large \frac{adjacent}{hypotenuse}$

tan x = $\large \frac{\sin x}{\cos x}=\frac{opposite}{adjacent}$

cot x = $\large \frac{1}{\tan x}$ = $\large \frac{adjacent}{opposite}$

sec x = $\large \frac{1}{\cos x}$ = $\large \frac{hypotenuse}{adjacent}$

cosec x/csc x = $\large \frac{1}{\sin x}$ = $\large \frac{hypotenuse}{opposite}$

Trigonometric Functions

Consider a unit circle with center at the origin.
Let P(a,b) be any point on the circle with ∠AOP = x radian, i.e. length of arc AP = x.
Consider the right angle triangle OMP,

$\sin x=a$

$\cos x=b$

$\tan x=\frac{b}{a}$

$\cot x=\frac{a}{b}$

$\sec x=\frac{1}{a}$

cosec x $=\frac{1}{b}$

Since OMP is a right angle triangle,
$\displaystyle \small OM^{2}$+ $\displaystyle \small MP^{2}$= $\displaystyle \small OP^{2}$
⇒ $\displaystyle \small a^{2}$+ $\displaystyle \small b^{2}$= 1
⇒ $\displaystyle \small \cos ^{2}x+\sin ^{2}x=1$

Values of sin x and cos x
We know that one complete revolution subtends an angle of 2π radian at the center of the circle.
If we start from A and move in anticlockwise direction, then at the points A,B,C,D and A, the arc length traveled are 0, π/2, π, 3π/2 and 2π.
Also the coordinates of points A,B,C and D are (1,0), (0,1), (-1,0) and (0,-1) respectively.
Therefore,

$\sin 0=0$
$\sin \frac{\pi }{2}=1$
$\sin \pi =0$
$\sin \frac{3\pi }{2}=-1$
$\sin 2\pi =0$

$\cos 0=1$
$\cos \frac{\pi }{2}=0$
$\cos \pi =-1$
$\cos \frac{3\pi }{2}=0$
$\cos 2\pi =1$

Further, sin x=0 when the point P on the unit circle coincides with the points A or C.
i.e. when x = 0, π, 2π, 3π,... or -π, -2π, -3π,...
i.e. when x = 0, ±π, ±2π, ±3π,... [when x is an integral multiple of π]
i.e. when x = nπ, where n is an integer.

Also cos x =0 when point P on the unit circle coincides with the points B and D.
i.e. when $x=\frac{\pi }{2},\frac{3\pi }{2},\frac{5\pi }{2},...or -\frac{\pi }{2},-\frac{3\pi }{2},-\frac{5\pi }{2},...$
i.e. when $x=\pm \frac{\pi }{2},\pm \frac{3\pi }{2},\pm \frac{5\pi }{2},...$ [when x is an odd multiple of $\frac{\pi }{2}$
i.e. when $x=\left ( 2n+1 \right )\frac{\pi }{2}$ , where n is an integer.

Thus,
$\sin x=0$ when $x=n\pi$ , n is any integer
$\cos x=0$ when $x=\left ( 2n+1 \right )\frac{\pi }{2}$ , n is any integer

Other Trigonometric Functions
The other trigonometric functions of the real number x are defined in terms of sine and cosine functions.

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

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

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

cot x = $\large \frac{\cos x}{\sin x},x\neq n\pi$ , n is any integer

Trigonometric Identities
(i) sin x cosec x = 1
(ii) cos x sec x = 1
(iii) tan x cot x = 1
(iv) sin(-x) = -sin x
(v) cos(-x) = cos x
(vi) tan(-x) = -tan x
(vii) cot(-x) = -cot x
(viii) sec(-x) = sec x
(ix) cosec (-x) = -cosec x
(x) $\displaystyle \small \cos ^{2}x+\sin ^{2}x=1$
(xi) $\displaystyle \small 1+\tan ^{2}x=\sec ^{2}x$
(xii) $\displaystyle \small 1+\cot ^{2}x=\csc ^{2}x$

Sign of Trigonometric Functions

Let O be the center of a unit circle.
Let P(a,b) be a point on the unit circle such that ∠AOP=x radian and length of arc AP=x.
We know that the circumference of the uint circle is 2π.
Also in this unit circle, -1≤a≤1 and -1≤b≤1.

In I quadrant, a>0, b>0, all functions are +ve.

In II quadrant, a<0, b>0, sin x and cosec x are +ve, other functions are -ve.

In III quadrant, a<0, b<0, tan x and cot x are +ve, other functions are -ve.

In IV quadrant, a>0, b<0, cos x and sec x are +ve, other functions are -ve.