Question

Suppose your graphing calculator has two functions, one called sin \(x,\) which calculates the sine of \(x\) when \(x\) is in radians, and the other called \(s(x),\) which calculates the sine of \(x\) when \(x\) is in degrees. a. Explain why \(s(x)=\sin \left(\frac{\pi}{180} x\right).\) b. Evaluate \(\lim _{x \rightarrow 0} \frac{s(x)}{x} .\) Verify your answer by estimating the limit on your calculator.

Short answer

a. The relationship between the sine function in radians and the sine function in degrees is given by \(s(x) = \sin(\frac{\pi}{180} x)\). b. The limit of the expression \(\frac{s(x)}{x}\) as \(x\) approaches 0 is \(\frac{\pi}{180}\).

Step-by-step solution

Relate radians and degrees in sine functions

To begin, recall that the sine function can be calculated in either radians or degrees, depending on the input value. The given function \(s(x)\) calculates the sine of \(x\) when \(x\) is in degrees. We need to find a relationship between \(s(x)\) and the sine function calculated in radians, which is represented by \(\sin{x}\). To relate radians and degrees, use the conversion factor: \(1 \text{ radian} = \frac{180}{\pi} \text{ degrees}\). Therefore, \(x\) radians is equal to \(\frac{180x}{\pi}\) degrees. Thus, we can rewrite the sine function in degrees as \(\sin{\frac{180x}{\pi}}\). Hence, \(s(x) = \sin(\frac{\pi}{180} x)\).

Find the limit of the given expression

Given the expression \(\frac{s(x)}{x}\), we now need to find the limit as \(x\) approaches 0. Substitute \(s(x)\) with the equivalent sine function in radians that we determined in Step 1: $$\lim_{x \to 0} \frac{s(x)}{x} = \lim_{x \to 0} \frac{\sin(\frac{\pi}{180} x)}{x}$$ To evaluate the limit, make a substitution: let \(u = \frac{\pi}{180}x\). Then, as \(x \rightarrow 0\), \(u \rightarrow 0\). Rewrite the expression in terms of \(u\): $$\lim_{u \to 0} \frac{\sin(u)}{\frac{180u}{\pi}}$$ Next, simplify the expression further: $$\lim_{u \to 0} \frac{\pi\sin(u)}{180u}$$ Now notice that we have reached a known limit property: $$\lim_{u \to 0} \frac{\sin(u)}{u} = 1$$ Therefore, the limit of the expression becomes: $$\lim_{x \to 0} \frac{s(x)}{x} = \frac{\pi}{180} \lim_{u \to 0} \frac{\sin(u)}{u} = \frac{\pi}{180} \cdot 1 = \frac{\pi}{180}$$

Verify the limit using a calculator

To confirm our result, we can estimate the limit by approaching 0 from both the left and the right using a graphing calculator. Calculate \(s(x)\) and divide by \(x\) for values very close to 0: $$\frac{s(0.001)}{0.001} \approx 0.0174533 \quad \text{and} \quad \frac{s(-0.001)}{-0.001} \approx 0.0174533$$ These values are approximately equal to the result we found, \(\frac{\pi}{180} \approx 0.0174533\). Thus, the limit is verified. The final answer for both parts is: a. \(s(x) = \sin(\frac{\pi}{180} x)\) b. \(\lim_{x \to 0} \frac{s(x)}{x} = \frac{\pi}{180}\)

Key concepts

Sine Function

The sine function is a fundamental concept in trigonometry, describing the relationship between the angles and side lengths in a right triangle. It's defined for all real numbers and commonly applied to angles measured in radians. The sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the hypotenuse of the triangle.
When dealing with periodic functions such as sine, it's essential to understand how it behaves both graphically and mathematically. The value of the sine function oscillates between -1 and 1, creating a wave-like shape when graphed.

• **Periodicity**: The sine function repeats its pattern every 2π radians or 360 degrees.
• **Amplitude**: The maximum value is 1 and the minimum value is -1, giving it an amplitude of 1.
• **Symmetry**: It's an odd function, meaning it's symmetric about the origin, expressed as sin(-x) = -sin(x).

Radians and Degrees Conversion

Understanding the conversion between radians and degrees is crucial for accurately handling trigonometric functions like sine. Degrees and radians are two units measuring angles, and depending on the context or device, you might need to switch between them.
The conversion factor between degrees and radians is based on the relationship that 180 degrees is equivalent to π radians. Thus, to convert from degrees to radians, you multiply by \(\frac{\pi}{180}\), and to convert back, you multiply by \(\frac{180}{\pi}\).
For example:

• 30 degrees = \(30 \times \frac{\pi}{180} = \frac{\pi}{6}\) radians.
• 1 radian = \(\frac{180}{\pi} \approx 57.2958\) degrees.

Using this conversion is particularly relevant for trigonometric calculators that compute sine in terms of degrees or radians. This method ensures that regardless of your starting measure, you can accurately translate it into the format needed for calculations.

Graphing Calculator

A graphing calculator can be an invaluable tool for visualizing mathematical concepts, especially when studying trigonometric functions such as the sine.These sophisticated calculators allow students to input complex equations and visualize their graphical representation.
Here's how a graphing calculator aids in understanding:

• **Visualization**: It helps plot functions like \(sin(x)\) over a specified range, making it easier to see its periodic nature.
• **Experimentation**: You can modify variables and see the effects in real-time, enhancing comprehension and interactive learning.
• **Verification**: As shown in the limit verification exercise, calculating for small values approaching zero can confirm mathematical results. This offers practical insights and adds confidence in your solutions.

To maximize its utility, users should ensure their calculator's settings match the desired unit (radians or degrees) and practice navigating the function menus efficiently.