Is a trig function with a constant in its parenthesis a constant?

Question

For example: Is $\sin(8)$ a constant? I want to know because my professor differentiated it to $0$ and that was the explanation he gave. Thanks

Answer 1

This is most definitely the case. It's true for any function not only trigonometric functions. When you plug a value into a function, you get one (and only one!) value, meaning it is a constant at that point. Think about the graph of your function. When you look at a specific value for the input variable (say $x=8$), the graph tells you what your function value is for that specific value.

Answer 2

Given any function $f(x)$ and a fixed value $x=x_1$, $f(x_1)$ is a constant.

Answer 3

Sin(8) is just a fancy way to write a fancy number. It is certainly constant.

If you take the derivative of e^2, 2π, or (42)², it is the same story. They are just numbers, it doesn't matter what symbols we used to write them.

Answer 4

Imagine you have two functions $f(x)=\sin{x}$ and $g(x)=8$, which the second is constant, then $$f(g(x))=\sin{8}$$ At that point you can see that $f(g(x))$ don't vary with $x$ then is constant. Other way to see it is via the chain rule: $$(f(g(x)))'=f'(g(x))g'(x)$$ What, for your case is: $$(f(g(x)))'=\cos(8)(8)'=cos(8)0=0$$ Then the function $f(g(x))=\sin{8}$ is constant for all $x$.

Hope it helps you to understand