Question

Does $\sin$ or $\cos$ start at $0$ .

Answer 1

To solve this question we should have the knowledge of working with functions. We have to use the concept of starting value of a function. Check the output value of both functions taking $0$ as input. After applying this concept we can have great results and we can get closer to the answer and can get the right answer. We should also have the knowledge of basic trigonometry.

Complete step by step answer:
For solving this question we should have the basic knowledge of function and concept of starting value of a function. We also have to use trigonometry and the concept of angles in trigonometry. Let us discuss one by one the concepts required to solve this question.
Function: A relation from a set of inputs to a set of outputs where each input is related to exactly one output. In a function we take input values and get output values for the same. Function is the base of entire calculus.
For e.g. $f(x)={{x}^{2}}+1$
Here, $f(x)$ is a function in terms of variable $x$ .
Starting value of a function: The starting value of the function is the output value of the function when the input value is $0$ . As we discussed earlier while talking about functions we have to take some input value and we get an output value as a result. In the concept of starting value of the function the input value is always $0$ and the output we get is known as the starting value of the function.
After knowing this entire concept let us try to solve the question.
In this question we have given two functions i.e. $\sin x$ and $\cos x$
The given functions fall in the category of trigonometric functions.
When $\sin x$ function is increasing $\cos x$ function is decreasing.
Applying the concept of starting value of a function,
We need to check the value of $f(x)$ at $x=0$ .
Here $x$ is the input
by concept of starting value of function $x=0$ and the output we get will be the starting value of the function
We need to check the output value.
If, output value $=0$
$0$ is the starting value of the function i.e. the given function starts at $0$
Checking the above concept on $\sin x$ function we get,
$\Rightarrow f(x)=\sin x$ ,
At $x=0$
$\Rightarrow$ $f(0)=\sin (0)$
By using basic trigonometry, we know $\sin (0)=0$
$\Rightarrow$ $f(0)=0$
Hence, the starting value of the function is $0$ .
We can conclude that $\sin x$ starts at $0$ .
Checking the above concept on $\cos x$ function we get,
$\Rightarrow f(x)=\cos x$ ,
At $x=0$
$\Rightarrow$ $f(0)=\cos (0)$
By using basic trigonometry, we know $\cos (0)=1$
$\Rightarrow$ $f(0)=1$
Hence, the starting value of the function is $1$ .
We can conclude that $\cos x$ starts at $1$ .
Hence, from $\sin$ or $\cos$ only $\sin$ starts at $0$ .

Note:
When $\sin x$ function is increasing $\cos x$ function is decreasing. Hence, we can conclude that if $\sin x$ function starts at $0$ , $\cos x$ function will definitely not start at $0$ . $\sin x$ and $\cos x$ are equal only when $x={{45}^{0}}$ in the first quadrant. We can use this same concept for other trigonometric functions. This can quickly and effectively solve these types of questions. $\sin x$ and $\cos x$ functions are dependent on each other.