Problem 36 - A Graphical Approach to Precalculus with Limits

Question

A student was asked to give an approximate value of sin \(45 .\) With her calculator in degree mode, she gave the value \(0.7071067812 .\) The teacher did not give her credit. What was her error?

Step-by-Step Solution

Answer

The student was not in error; her calculator value was correct for sin(45°), but radians were expected.

Step 1: Convert Degrees to Radians

First, we need to ensure that the angle is given in radians since trigonometric functions in calculus and higher mathematics often require radians. Use the conversion formula: \[ \text{Radians} = \frac{\pi}{180} \times \text{Degrees} \]So, for an angle of 45 degrees,\[ \text{Radians} = \frac{\pi}{180} \times 45 = \frac{\pi}{4} \]

Step 2: Calculate Sin in Radians

Now, let's find \(\sin\left(\frac{\pi}{4}\right)\), which is the sine of 45 degrees in radians. By definition, for 45 degrees: \[ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \approx 0.7071 \]This is the correct value when considering the angle in radians.

Step 3: Verify Calculator Mode

The student obtained the correct numerical result, \(0.7071067812\), for \(\sin(45^\circ)\), assuming it meant \(\sin(45)\) evaluated in degrees. However, in a context where radians are expected, the student seemed ignorant of the calculator mode. Ensure the calculator is in the correct mode (radians if radians are required).

Key Concepts

Radians and Degrees

Understanding radians and degrees is crucial in trigonometry because they are the two units used to measure angles. Degrees are more commonly used in everyday situations, while radians are preferred in more advanced fields like calculus. This is because radians relate the angle to the radius of a circle, making calculations involving trigonometric functions more natural and straightforward. When converting degrees to radians, use the formula:
\[\text{Radians} = \frac{\pi}{180} \times \text{Degrees}\]
For example, to convert 45 degrees to radians, plug it into the formula to get:

• \[\text{Radians} = \frac{\pi}{180} \times 45 = \frac{\pi}{4}\]

Using radians allows for simpler integration and differentiation in calculus, making it essential to be comfortable with the conversion and usage of radians alongside degrees.

Sine Function

The sine function is one of the basic trigonometric functions that relate the angle of a right triangle to the length of the side opposite the angle divided by the hypotenuse. In more advanced mathematics, the sine function is periodic and can be defined for all real numbers via the unit circle concept or its series expansion.

The value of the sine function often used in basic trigonometry and remembered as:

• sin(0) = 0
• sin(30°) = 0.5
• sin(45°) = \(\frac{\sqrt{2}}{2} \approx 0.7071\)
• sin(60°) = \(\frac{\sqrt{3}}{2}\)
• sin(90°) = 1

For the problem at hand, recognizing the sine value for 45 degrees as \(\frac{\sqrt{2}}{2}\) in degrees and the same when evalued as \(\sin\left(\frac{\pi}{4}\right)\) in radians shows the importance of evaluating angles correctly in their respective units.

Calculator Mode Error

One common error when working with angles and trigonometric functions is a calculator mode error. Calculators can operate in different modes, such as degrees and radians. Using the wrong mode can lead to correct numerical results, but incorrect interpretation in math problems.

To avoid this error, always check the calculator's mode before beginning calculations. If radians are expected for your problem, make sure the calculator is set to radians. Similarly, if degrees are required, set it to degrees. This small step ensures consistent results and prevents confusion, as seen in the student's error when calculating \(\sin(45^\circ)\) and obtaining the correct numerical value, but without applying the appropriate mode understanding in the calculus context.

• Always confirm the mode: look for 'DEG' or 'RAD' on your calculator's display.
• Change modes if necessary before calculating trigonometric functions.
• Recheck especially during exams or complex problem solving.

Following these practices can save time and ensure accuracy in math calculations.