Maths Online by SHAH HUSSAIN
The secant and cosecant have periods of length 2π, and we don't consider amplitude for these curves. The cotangent has a period of π, and we don't bother with the amplitude. ... If you know the behavior of the function at zero, π/2, π, 3π/2, and 2π, then you can fill in the rest.
▪️The cosecant is the reciprocal of the sine. Wherever the sine is zero, the cosecant will be undefined, so there will be a vertical asymptote. Wherever the sine reaches its maximum value of 1, the cosecant will reach its minimum value of 1; wherever the sine reaches its minimum value of –1, the cosecant will reach its maximum value of –1. Wherever the sine is positive but less than 1, the cosecant will be positive but greater than 1; wherever the sine is negative but greater than –1, the cosecant will be negative but less than –1.cotangent is the reciprocal of the tangent. Wherever the tangent is zero, the cotangent will have a vertical asymptote; wherever the tangent has a vertical asymptote, the cotangent will have a zero. And the signs on each interval will be the same.
☆PERIOD OF TRIGONOMETRIC FUNCTION:
The period of trigonometric function is the smallest positive number which, when added to the original circular measure of angle, give the same value of the function therefore sine function is a periodic function and its period is 2π .
☆DOMAIN :
As cosecx is not defined at even multiples of 90° therefore Domain of cosecx is set of all r eal numbers except even multiples of 90°
i.e
Domain of cosecx= R-{nπ} where n € Z
☆RANG:
Range of cosecx = (- infinity , -1] U [1, infinity)
☆FREQUENCY:
As frequency is the reciprocal of period and period of cosecx is 2π so its frequency is 1/2π
☆AMPLITUDE:
The amplitude is the height from the centre line to the peak.
As range of cosecx=(infinity , -1] U [1, infinity) therefore amplitude does not defined
☆PHASE SHIFT:
The phase shift is how far the function is horizontally to the right of the usual position.
As there is no change in cosecx angle so its phase shift is zero
☆GRAPH:
As all trigonometric functions are periodic so graph of sinx within the domain is smooth curve.
☆SYMMERTRY:
If function is an even function then its graph's symmetry about y -axis
and if functions is odd then its graph's symmetry about origin. So
cosecx is an odd function therefore it is symmetry about y-axis
☆NATURE OF FUNCTION:
cosecx is an odd function by nature
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