Inverse Circular Functions
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INVERSE CIRCULAR FUNCTIONS
In trigonometry the inverse circular or the inverse trigonometric functions are
- sin-1x
- cos-1x
- tan-1x
- cot-1x
- cosec-1x
- sec-1x.
Sin-1x, gives the angle for which sin angle is x.
For example, sin 30‚ = 1/2
è sin-1(1/2) = 30
Similarly for other trigonometric functions.
RANGE
More than one angle has the same value of the sine ratio:
sin 30 = sin 150 = sin 390 = . . .
In order to make the value of sin-1(x) unique, its value should be an angle in the interval [-90‚ 90] or [-p/2, p/2]. Similarly functions have to tie in a specific interval, called the ‘range’ of the function. The ranges of all the functions are:
- sin-1x
x is an element of [-1,1]
therefore range = [-p/2, p/2]
- cos-1x
x is an element of [-1, 1]
Therefore range [0, p]
- tan-1x
x is an element of (-P, P)
Therefore range (-p/2, p/2)
- cot-1x
x is an element of (-P, P)
Therefore range: (0, p)
Thus, when x is positive sin-1x belongs to (0, p/2), and when x is negative sin-1x belongs to (-p/2, 0)
SOME BASIC EQUALITIES
When x > 0 sin-1x = cos-1(1-x2)
When x <0, sin-1x = -cos-1(1-x2)
when x > 0 cos-1x = sin-1(1-x2)
when x < 0 cos-1x = p - sin-1(1-x2)
when x > 0 tan-1x = cot-1(1/x)
when x < 0 tan-1x = cot-1 (1/x) - p
when x > 0 cos-1x = tan-12(1-x2) / x
when x < 0 cos-1x = p + tan-12(1-x2) / x
when x,y > 0, xy <1
tan-1x + tan-1y = {tan-1 {(x+y)/(1-xy)}
when x,y > 0, xy > 1
{tan-1 {(x+y)/(1-xy)} + p
when x < 0, xy > 1
{tan-1 {(x+y)/(1-xy)} - p