Question

If \[x\] is measured in degrees, 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

Hint: First we have to convert degree to radian. Derivative of a trigonometric function in degree does not exist. Then we will apply the formula \[\dfrac{d}{{dx}}\left( {\cos x} \right) = - \sin x\]

Formula Used: To the covert degree to radian, we have the formula \[{180^ \circ } = {\pi ^c}\]. \[\dfrac{d}{{dx}}(\cos x) = - \sin x\], where x is measured in radian. We can write \[\dfrac{{dy}}{{dx}} = \dfrac{{dy}}{{dz}}.\dfrac{{dz}}{{dx}}\] where y is a function of z and z is a function of x. This is called Chain Rule. Using this rule, we have \[\dfrac{d}{{dx}}(\cos y) = \dfrac{d}{{dy}}(\cos y)\dfrac{{dy}}{{dx}}\]

Complete step-by-step solution:
Here, x is measured in degrees. First, we have to convert the degree to radian. Derivative of trigonometric function in degree does not exist. Only the derivative of the trigonometric function in radian exists.
Therefore, \[{180^ \circ } = {\pi ^c}\]
or, \[{1^ \circ } = \dfrac{{{\pi ^c}}}{{180}}\]
or, \[{x^ \circ } = \dfrac{\pi }{{180}}{x^c}\]
Therefore,
\[\dfrac{d}{{dx}}(\cos {x^ \circ }) = \dfrac{d}{{dx}}(\cos \dfrac{{\pi {x^c}}}{{180}})\]
Let \[y = \dfrac{{\pi x}}{{180}}\]
\[\dfrac{d}{{dx}}(\cos y) = \dfrac{d}{{dy}}(\cos y)\dfrac{{dy}}{{dx}}\] [Using Chain Rule]
\[ = ( - \sin y)\dfrac{d}{{dx}}(\dfrac{{\pi x}}{{180}})\]
\[ = - \sin y \times \dfrac{\pi }{{180}} \times \dfrac{d}{{dx}}(x)\]
\[ = - \sin y \times \dfrac{\pi }{{180}} \times 1\]
\[ = \dfrac{\pi }{{180}} \times - \sin y\]
\[ = - \dfrac{\pi }{{180}}\sin (\dfrac{{\pi x}}{{180}})\]

Hence, option C. is correct.

Note: Students often mistake not to convert degree to radian. They only take derivatives and find the relevant option to choose the correct answer. They do not check whether it is in degree form or radian form. So, after getting the question, you have to notice carefully the question that is given in either degree form or radian form. If it is in radian form then you don’t convert. But if it is in degree form then you must convert it to radian form using the above formula and then take the derivative using the Chain Rule. If there is not mentioned anything in question then you should know that it is in radian form.