Question

True or False \(\frac{d}{d x}(\sin x)=\cos x,\) if \(x\) is measured in degrees or radians. Justify your answer.

Short answer

False. The derivative of \( \sin x \) with respect to \( x \) is \( \cos x \) when \( x \) is measured in radians, and \( \frac{\pi}{180} \cos x \) when \( x \) is measured in degrees.

Step-by-step solution

Understand the derivative in radians

The derivative of \( \sin x \) with respect to \( x \) is \( \cos x \) when \( x \) is measured in radians. This is a fundamental rule in calculus. This can be proven using limits definition of derivatives.

Understand the derivative in degrees

When \( x \) is measured in degrees, the situation changes. Degrees and radians are just different units for measuring angles, like inches and centimeters are different units for measuring length. There's a conversion factor between the two, specifically, \( \pi \) radians is the same as 180 degrees. Therefore, when we're taking the derivative of \( \sin x \) where \( x \) is measured in degrees, it's like we're measuring in radians but with a conversion factor. This results in the derivative being \( \cos x \) times the conversion factor from degrees to radians, which is \( \frac{\pi}{180} \). This is because the Chain rule in derivatives states that if you have a function inside another function, the derivative is the derivative of the outer function times the derivative of the inner function.

Conclusion

Therefore, the derivative of \( \sin x \) with respect to \( x \) is not just \( \cos x \) when \( x \) is measured in degrees. It's \( \frac{\pi}{180} \cos x \). So, the statement is false when \( x \) is measured in degrees.