Question

If $x$ is measured in degree, then $\dfrac{d}{{dx}}(\cos x)$ is equal to:
A: $- \sin x$
B: $- \dfrac{{180}}{\pi }\sin x$
C: $- \dfrac{\pi }{{180}}\sin x$
D: $\sin x$

Answer 1

Usually whenever it comes to trigonometric functions the $x$ will be measured in terms of radians. So to convert radians to degrees we have a formula, which is given by: $x\deg ree = \dfrac{\pi }{{180}}xradians$. Now in place of $x$ substitute the value in terms of degree and solve for the correct answer.

Complete Step by Step Solution:
In the question they have given that $x$ is measured in terms of degree.
They have asked to find the derivative of $\cos x$ in terms of degree.
Usually in trigonometric functions, the value of $x$ will be given in terms of radians.
Here they are asking in terms of degree, so to convert radians to degree we have a relation between degree and radians given by:
$x\deg ree = \dfrac{\pi }{{180}}\text{x radians}$
By using the above relation we need to find the required answer.
Given function is $\dfrac{d}{{dx}}(\cos x)$
Now, replace $x$ with $\dfrac{\pi }{{180}}x$ in the above function, we get
$\dfrac{d}{{dx}}(\cos x) = \dfrac{d}{{dx}}\left( {\cos \dfrac{{\pi .x}}{{180}}} \right)$
Now, we know that differentiation of $\cos x$ is $- \sin x$.so the above equation becomes,
$\dfrac{d}{{dx}}\left( {\cos \dfrac{{\pi .x}}{{180}}} \right) = - \sin x\dfrac{d}{{dx}}\left( {\dfrac{\pi }{{180}}x} \right)$

$\Rightarrow - \sin x\dfrac{d}{{dx}}\left( {\dfrac{\pi }{{180}}x} \right) = - \dfrac{\pi }{{180}}\sin x$

Therefore when $x$ is measured in terms of degree then the derivative of $\cos x$ will be $- \dfrac{\pi }{{180}}\sin x$. Hence option C is the correct answer.

Note:
In this conversion type problem one thing we need to remember is the conversion formula. In case of trigonometric functions we need to remember the derivative of all the functions, because they may ask for other trigonometric functions instead of $\cos x$. Both conversion formula and derivative of trigonometric functions are important, if any one of these goes wrong while solving the problem then it may lead to the incorrect solution.