Problem 58 - Thomas Calculus

Question

Radians versus degrees: degree mode derivatives What happens to the derivatives of \(\sin x\) and \(\cos x\) if \(x\) is measured in degrees instead of radians? To find out, take the following steps. a. With your graphing calculator or computer grapher in degree mode, graph $$ f(h)=\frac{\sin h}{h} $$ and estimate \(\lim _{h \rightarrow 0} f(h) .\) Compare your estimate with \(\pi / 180 .\) Is there any reason to believe the limit should be \(\pi / 180 ?\) b. With your grapher still in degree mode, estimate $$ \lim _{h \rightarrow 0} \frac{\cos h-1}{h} $$ c. Now go back to the derivation of the formula for the derivative of \(\sin x\) in the text and carry out the steps of the derivation using degree-mode limits. What formula do you obtain for the derivative? d. Work through the derivation of the formula for the derivative of cos \(x\) using degree-mode limits. What formula do you obtain for the derivative? e. The disadvantages of the degree-mode formulas become apparent as you start taking derivatives of higher order. Try it. What are the second and third degree-mode derivatives of \(\sin x\) and \(\cos x ?\)

Step-by-Step Solution

Answer

Derivatives in degrees involve a conversion factor of \( \frac{\pi}{180} \). Higher order derivatives accumulate this factor.

Step 1: Evaluate Limit of Sine Ratio

First, consider the expression \( f(h) = \frac{\sin h}{h} \) as \( h \to 0 \) with \( h \) in degrees. In radians, we know \( \lim_{h \to 0} \frac{\sin h}{h} = 1 \). In degrees, however, \( \sin(1^\circ) \approx \frac{\pi}{180} \), hence \( \lim_{h \to 0} \frac{\sin h}{h} \approx \frac{\pi}{180} \). This suggests that the limit should be \( \pi/180 \).

Step 2: Evaluate Cosine Limit

Now, evaluate \( \lim_{h \to 0} \frac{\cos h - 1}{h} \) with \( h \) in degrees. In radians, this limit is \( 0 \). Applying a similar modification as in Step 1, convert the radians approximation to degrees, which results in \( 0 \times \frac{\pi}{180} = 0 \).

Step 3: Derivative of Sine in Degrees

Re-derive the derivative of \( \sin x \) using degree-based limits. Begin with \( \frac{d}{dx} \sin x = \lim_{h \to 0} \frac{\sin(x+h) - \sin x}{h} \). This becomes \( \cos(x) \cdot \frac{\pi}{180} \), a key observation showing that the conversion factor for derivatives is \( \frac{\pi}{180} \).

Step 4: Derivative of Cosine in Degrees

Similarly, derive the derivative of \( \cos x \) using degree limits. Starting with \( \frac{d}{dx} \cos x = \lim_{h \to 0} \frac{\cos(x+h) - \cos x}{h} \), yields \( -\sin(x) \cdot \frac{\pi}{180} \). Again, the degree-radian conversion factor \( \frac{\pi}{180} \) appears in the formula.

Step 5: Higher-Order Derivatives

Apply the degree-mode to find second and third derivatives. For \( \sin x \), the second derivative is \( -\sin x \cdot \left( \frac{\pi}{180} \right)^2 \), and the third derivative is \( -\cos x \cdot \left( \frac{\pi}{180} \right)^3 \). Similarly for \( \cos x \): the second derivative is \( -\cos x \cdot \left( \frac{\pi}{180} \right)^2 \), and the third derivative is \( \sin x \cdot \left( \frac{\pi}{180} \right)^3 \).

Key Concepts

Understanding Radian vs Degree Measurement

When dealing with trigonometric functions and their derivatives, it's crucial to understand how radian and degree measurements influence the results. In mathematics, radians are generally preferred over degrees for several reasons. Primarily, the use of radians simplifies the computation of derivatives. This is because the radian measure ties directly into the unit circle notion, where the length of the arc on the unit circle is considered. This allows the trigonometric limits to simplify naturally.

• A radian is the angle made when the arc length is equal to the radius of the circle.
• Degrees measure angles with a full circle having 360 degrees.
• Conversion between degrees and radians: 180 degrees equals π radians, which introduces a conversion factor of \( \frac{\pi}{180} \) when switching from radians to degrees.

Switching between these two units affects how derivatives are computed, as is evident in trigonometric limits, where radians yield neat results like \( \lim_{h \to 0} \frac{\sin h}{h} = 1 \). In contrast, working in degrees necessitates adjustments by the factor of \( \frac{\pi}{180} \). Understanding this conversion is vital for correctly interpreting trigonometric derivatives when changing from radians to degrees.

Limits in Trigonometric Functions

Limits are fundamental in calculus and especially important when dealing with trigonometric functions. One often encounters two classic limits involving sine and cosine functions. These are:

• \( \lim_{h \to 0} \frac{\sin h}{h} = 1 \) in radians.
• \( \lim_{h \to 0} \frac{\cos h - 1}{h} = 0 \) in radians.

These limits change when you use degrees instead of radians due to the linear scaling of degrees to radians with \( \frac{\pi}{180} \). In degree mode:

• \( \lim_{h \to 0} \frac{\sin h}{h} \approx \frac{\pi}{180} \).
• \( \lim_{h \to 0} \frac{\cos h - 1}{h} = 0 \), as the scaling does not change the concept that cosine approaches zero change.

Understanding these conversions is critical, as misapplying them may lead to incorrect results when calculating derivatives or evaluating trigonometric limits. It highlights why radians are often used as the standard unit for calculus, creating smooth transitions and clear expressions without needing additional conversion factors.

Exploring Higher Order Derivatives

When calculating derivatives of higher order for trigonometric functions, the conversion factor from degrees to radians becomes increasingly important. For functions like \( \sin x \) and \( \cos x \), the higher order derivatives follow a cyclical pattern.For degree-mode derivatives, this cyclical nature is impacted by the repeated application of the radian conversion factor.

• The second derivative of \( \sin x \) is \( -\sin x \cdot \left( \frac{\pi}{180} \right)^2 \), and the third derivative is \( -\cos x \cdot \left( \frac{\pi}{180} \right)^3 \).
• The second derivative of \( \cos x \) is \( -\cos x \cdot \left( \frac{\pi}{180} \right)^2 \), and the third derivative is \( \sin x \cdot \left( \frac{\pi}{180} \right)^3 \).

The regular appearance of \( \left( \frac{\pi}{180} \right)^n \) as the order of derivative increases emphasizes the suppression of derivative magnitudes, showing how degree measurements significantly alter derivative behavior. This repeated scaling impacts the amplitude and makes derivatives in degrees seemingly inconvenient for practical purposes.

Understanding Sine and Cosine Derivatives

The derivatives of the sine and cosine functions are foundational in calculus. When measured in radians, these derivatives are simple and symmetric, but they require adjustments when measured in degrees due to the conversion factor \( \frac{\pi}{180} \).

• The derivative of \( \sin x \) in degree mode is \( \cos(x) \cdot \frac{\pi}{180} \). This indicates that as we differentiate, the conversion factor suppresses the derivative's magnitude.
• The derivative of \( \cos x \) becomes \( -\sin(x) \cdot \frac{\pi}{180} \).

The inclusion of the \( \frac{\pi}{180} \) factor means that in degree mode, derivatives tend to be smaller in magnitude compared to their radian counterparts. This factor reflects the linear conversion between angles in degrees and radians, affecting the rate of change calculated by the derivatives, thus portraying a realistic scenario of how angle measurement systems impact trigonometric analysis in calculus.