Problem 35 - Precalculus

Question

Use a calculator to evaluate the trigonometric functions for the indicated angle values. Round your answers to four decimal places. $$\sin \left(\frac{\pi}{12}\right)$$

Step-by-Step Solution

Answer

\( \sin \left(\frac{\pi}{12}\right) \approx 0.2588 \)

Step 1: Convert the Angle to Degrees

The angle given is \( \frac{\pi}{12} \) radians. To convert it to degrees, use the conversion factor \( \frac{180}{\pi} \). So, \( \frac{\pi}{12} \times \frac{180}{\pi} = 15^{\circ} \). Therefore, the angle is 15 degrees.

Step 2: Use Calculator to Find the Sine Value

Set your calculator to degree mode. Enter 15 and then use the sine function to calculate \( \sin(15^{\circ}) \).

Step 3: Round the Result

The calculator should give you a result like 0.2588190451. Round this result to four decimal places to get \( 0.2588 \).

Key Concepts

Radians to Degrees Conversion

When working in trigonometry, angles can be expressed in two different units: radians and degrees. It's important to understand how to convert between them.
Radians are an angular measurement based on the radius of a circle. One complete rotation around a circle is equal to \(2\pi\) radians.
Degrees are more common in everyday use, where a full circle is divided into 360 equal parts, or degrees.

To convert an angle from radians to degrees, use the conversion factor \( \frac{180}{\pi} \).
Simply multiply the radian measure by this factor.

For example, to convert \( \frac{\pi}{12} \) radians to degrees:

• Multiply by \( \frac{180}{\pi} \)
• This gives \( \frac{\pi}{12} \times \frac{180}{\pi} = 15^{\circ} \)

This straightforward calculation helps in switching from one unit to the other, crucial when preparing to use a calculator for certain functions.

Sine Function

The sine function is one of the fundamental trigonometric functions. It is especially useful in calculating relationships within right triangles and periodic phenomena, like waves.

When you see \( \sin \theta \), it represents the ratio of the length of the opposite side to the hypotenuse in a right triangle.

In the context of a unit circle, the sine of an angle is equal to the y-coordinate of the point on the circle that corresponds to the angle.

For example, in the unit circle, the sine of 15 degrees (\( \sin(15^{\circ}) \)) can be calculated using a calculator because it's not one of the standard angles typically memorized in trigonometry.
Using the calculator, you enter the angle and get the sine value. For 15 degrees, this calculation provides a result of approximately \( 0.2588190451 \), which is typically rounded during solutions.

Calculator Usage

A calculator is a powerful tool in solving trigonometric problems. For calculating functions like sine, precision and understanding the tool's settings are key.

First, ensure your calculator is set to the correct mode. For our example using degrees, switch the calculator to degree mode before entering the angle.
This setting tells the calculator how to interpret the input angle. If left in radian mode, the result will be incorrect for degrees.

Next, input the angle directly. For instance, enter 15 for 15 degrees and press the sine function button. Most scientific calculators have a dedicated button for \( \sin \).
This process yields the sine of the angle by providing a result like \( 0.2588190451 \).

Finally, rounding to the desired precision, typically four decimal places, results in \( 0.2588 \). This ensures clarity and consistency in mathematical and practical applications.