How do you find the domain and range of sine, cosine, and tangent?
1 Answer
The domain and range of trigonometric functions are determined directly from the definition of these functions.
Let's start from the definition.
Trigonometric functions are defined using a unit circle on a coordinate plane - a circle of a radius
Consider a point
The value of this angle can be positive (if we go counterclockwise from
Each value of an angle from
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Function
y=sin(x) is defined as the ordinate (Y -coordinate) of a point on a unit circle that corresponds to an angle ofx radians. Therefore, the domain of this function is all real numbers from-oo to+oo . The range is from-1 to+1 since this is an ordinate of a point on a unit circle. -
Function
y=cos(x) is defined as the abscissa (X -coordinate) of a point on a unit circle that corresponds to an angle ofx radians. Therefore, the domain of this function is all real numbers from-oo to+oo . The range is from-1 to+1 since this is an abscissa of a point on a unit circle. -
Function
y=tan(x) is defined assin(x)/cos(x) . The domain of this function is all real numbers except those wherecos(x)=0 , that is all angles except those that correspond to points(0,1) and(0,-1) . These angles wherey=tan(x) is undefined arepi/2 + pi*N radians, whereN - any integer number. The range is, obviously, all real numbers from-oo to+oo .
Of special interest might be the graphs of these functions. You can refer to a series of lectures on Unizor dedicated to detailed analysis of these functions, their graphs and behavior.