Problem 16 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Why have we been telling you that radians are more appropriate than degrees when using calculus? Suppose \(x\) is measured in degrees. Then \(\cos x^{\circ}=\cos \left(\frac{x^{\circ} \pi \text { radians }}{180^{\circ}}\right)=\) \(\cos \left(\frac{\pi x}{180}\right)\) where the argument is now in radians. Find the derivative. Is the derivative \(-\sin x^{\circ} ?\)

Step-by-Step Solution

Answer

No, the derivative of \(\cos x^{\circ}\) is \(-\sin\left(\frac{\pi x}{180}\right) \cdot \frac{\pi}{180},\) not \(-\sin x^{\circ}\). The presence of the extra factor of \(\frac{\pi}{180}\) highlights one reason why calculations in calculus are frequently performed in radians instead of degrees.

Step 1: Write Down the Function

Firstly, rewrite the given function with \(x\) denoting degrees. Thus, \(f(x) = \cos \left(\frac{\pi x}{180}\right)\).

Step 2: Apply the Chain Rule

The chain rule states that if \(g(x) = f(u(x))\), then \(g'(x) = f'(u(x)) \cdot u'(x)\). Apply the chain rule to \(f(x)\). The outer function \(f(u) = \cos(u)\) has derivative \(-\sin(u)\) and the inner function \(u(x) = \frac{\pi x}{180}\) has derivative \(\frac{\pi}{180}\). According to the Chain rule, this would mean \(f'(x) = -\sin(u) \cdot u'(x) = -\sin\left(\frac{\pi x}{180}\right) \cdot \frac{\pi}{180}\)

Step 3: Compare with the Derivative of the Sin Function

The derivative of \(-\sin x^{\circ}\) in radians is \(-\cos x\). However, notice that our result from last step has an additional factor of \(\frac{\pi}{180}\) when compared to the derivative of sin function. Thus, the derivative does not equal \(-\sin x^{\circ}\).

Key Concepts

Radians

Radians are a way of measuring angles that is often more suitable for calculus than degrees. This is because radians have a natural relationship with arc lengths and the unit circle. In radians, the angle measure directly corresponds to the arc length on the unit circle. This makes mathematical calculations simpler and more intuitive. By using radians, many trigonometric function derivatives become straightforward and elegant expressions. For instance, the derivative of \( \sin(x) \) in radians is simply \( \cos(x) \), while using degrees would involve additional conversion factors.
This is because 360 degrees is equivalent to \( 2\pi \) radians, embedding the circle's circumference into the radian measure. When calculating derivatives in calculus, using any angle involves multiplying by constant conversion factors if degrees are used. Radians remove this extra step, directly integrating into the operation and providing cleaner results.

Chain Rule

The Chain Rule is a powerful tool in calculus for differentiating composite functions. If you have a function that can be expressed as another function applied to a variable, the Chain Rule helps you find its derivative. Mathematically, if \( g(x) = f(u(x)) \), then the derivative \( g'(x) = f'(u(x)) \cdot u'(x) \).
This means you first find the derivative of the outer function \( f \) with respect to its input \( u \), and then multiply it by the derivative of the inner function \( u \) with respect to \( x \). In our example, the function is \( \cos(\frac{\pi x}{180}) \).

• The outer function is \( \cos(u) \), whose derivative is \( -\sin(u) \).
• The inner function is \( \frac{\pi x}{180} \), whose derivative is \( \frac{\pi}{180} \).

By applying the Chain Rule, we get the result \( -\sin\left(\frac{\pi x}{180}\right) \cdot \frac{\pi}{180} \), which highlights the crucial role of this rule in differentiating nested functions.

Trigonometric Derivatives

Trigonometric functions are fundamental in calculus, and understanding their derivatives is key to solving many problems. The most common trigonometric derivatives are:

• The derivative of \( \sin(x) \) is \( \cos(x) \).
• The derivative of \( \cos(x) \) is \( -\sin(x) \).

When working with these derivatives in degrees rather than radians, additional factors emerge. In the given exercise, \( \cos(x^\circ) \) was converted to \( \cos(\frac{\pi x}{180}) \) to take advantage of the radian measure.
This transformation allows for straightforward derivative calculation using known rules. Without this conversion, derivatives taken directly in degrees would result in expressions with unwanted scaling factors. As seen, the derivative is affected by a factor of \( \frac{\pi}{180} \), confirming that using radians is more natural and efficient for calculus operations involving trigonometric functions.