Find range and domain of f(x)=1/(0.5-sinx)
Answers
Domain and Range of Sin(x)
To find the domain and range of sin(x) let's imagine a circle with radius 1 and center at the origin. For any point on this circle, if we draw a right-angled triangle its hypotenuse will always be 1. Like this:
sine
As we understand, the sin(x) is defined as the opposite divided by the hypotenuse. For this unit circle, at any point, sin(x) is equal to opposite / 1. This measure of opposite can be defined for all the points on the circle, indicating that the angle x can take any value. So, the domain of sin(x) is all real numbers.
Also, the value of sin(x), depending on the point on the circle, can go to a maximum of 1 at x = 90 degrees and a minimum of -1 at x = 270 degrees. So, the range of sin(x) is -1 to 1.
In short, for y = sin(x):
Domain = [+ ∞, - ∞]
Range = [-1, +1]
The function y = sin(x) can be plotted as shown in this image:
Graph of sin(x)
The domain and range of csc(x) can be calculated as follows: the csc(x) is the reciprocal of sin(x). Its domain and range can be found from the domain and range of sin(x). The csc(x) cannot be defined for those values of x for which sin(x) = 0. Its domain is all real numbers excluding x = 0 degrees, 180 degrees, 360 degrees, and so on. Similarly, as the value of the range of sin(x) lies between -1 to 1, the value of its reciprocal is either greater than or equal to 1 or lesser than or equal to -1.
Therefore, for the function y = csc(x):
Domain = R - nπ
Range = all values that belong to the set [- ∞, - 1] ∪ [+ 1, + ∞]
The domain and range of sine inverse is defined as:
y = sin-1 x
which means:
x = sin(y)
In short, the inverse function of sin(x) is defined for all the points that correspond to a sin(x) value, which means that its domain is equal to the range of sin(x), -1 to 1. The range of this inverse function is the angles where sin(x) has a defined value, from -90 degrees to 90 degrees, as these values cover all the outputs of sin(x).
For y = sin-1 x
Domain = [ +1, - 1]
Range = [- π/2, + π/2]