Question

How do you find the domain and range of sine, cosine, and tangent?

Answer 1

Hint: The sine, cosine, and tangent functions are important in trigonometry and many other areas of mathematics. The definition of sine, cosine, and tangent can be extended to the complex numbers by defining the functions by their Taylor series instead of by the ratio of two lengths.
 First, we will design the graph for them. And based on the graphs, we will find the domain and range of them.

Complete step-by-step answer:
Domain and range of sine function, $y = \sin \left( x \right)$:

There is no restriction on the domain of sine function.
Therefore, the domain of sine function is $x \in R$.
The range of sine function is -1 to 1.
Therefore, we can say that the domain and range of sine function is all complex numbers.
Domain and range of cosine function, $y = \cos \left( x \right)$:

There is no restriction on the domain of cosine function.
Therefore, the domain of cosine function is $x \in R$.
The range of sine function is -1 to 1.
Therefore, we can say that the domain and range of cosine function is all a complex number.
Domain and range of tangent function, $y = \tan \left( x \right)$:

The tangent function has vertical asymptotes at $ \pm \dfrac{{\left( {2n + 1} \right)\pi }}{2}$.
Therefore, the domain of the tangent function is $x \ne \pm \dfrac{{\left( {2n + 1} \right)\pi }}{2}$.
So, the range of the tangent function is $y \in R$.

Therefore, we can say that the domain of the tangent function is all complex numbers expect $ \pm \dfrac{\pi }{2}, \pm \dfrac{{3\pi }}{2}, \pm \dfrac{{5\pi }}{2},...$ , where the tangent function is undefined, the range is all complex numbers.

Note:
Domain: The domain of a function is the specific set of values that the independent variable in a function can take on.
Range: The range is the resulting value that the dependent variable can have as x varies throughout the domain.