Problem 36 - Precalculus

Question

Use a calculator to evaluate the trigonometric functions for the indicated angle values. Round your answers to four decimal places. $$\sin \left(\frac{5 \pi}{9}\right)$$

Step-by-Step Solution

Answer

\( \sin\left(\frac{5\pi}{9}\right) \approx 0.9848 \).

Step 1: Convert Radians to Degrees

First, convert the angle from radians to degrees. Use the formula: degrees = radians \( \times \frac{180}{\pi} \). For this problem, the angle in radians is \( \frac{5\pi}{9} \). Convert it to degrees: \( \frac{5\pi}{9} \times \frac{180}{\pi} = 100 \) degrees.

Step 2: Find Sine of the Angle

With the angle in degrees (100 degrees), use a calculator to find the sine. Set the calculator to degree mode (if not already set), then enter 100 and press the sine function. The sine of 100 degrees is approximately 0.9848 when rounded to four decimal places.

Key Concepts

Radians to Degrees Conversion

Converting between radians and degrees is essential in trigonometry because these are the two main units of angle measurement. Radians are typically used in higher mathematics, while degrees are more common in everyday contexts. To convert from radians to degrees, you use the formula:

• degrees = radians \( \times \frac{180}{\pi} \)

This formula comes from the fact that a full circle is both \( 2\pi \) radians and 360 degrees. Therefore, \( \pi \) radians equal 180 degrees.
To convert \( \frac{5\pi}{9} \) radians to degrees:

• Multiply \( \frac{5\pi}{9} \) by \( \frac{180}{\pi} \).
• The \( \pi \) symbol cancels out, simplifying your calculations.
• You'll find that \( \frac{5\pi}{9} \times \frac{180}{\pi} = 100 \) degrees.

Remember, converting radians to degrees helps make it easier to understand and work with angles when performing trigonometric calculations or using a calculator.

Sine Function Calculation

The sine function is one of the primary trigonometric functions and is used to relate an angle to the ratios of the sides in a right triangle. When working with these angles, especially if derived from radians, it's crucial to have the correct unit in mind.
After converting your angle from radians to degrees, which is 100 degrees in this exercise, you can find the sine using a calculator. Here's how:

• Ensure your calculator is set to degree mode. This setting is vital because if it is in radian mode, the result will be incorrect.
• Type in the angle, 100 in this case, and then press the sine (sin) function key.
• Your result will be the sine of 100 degrees, approximately 0.9848, once rounded to four decimal places.

Understanding how to find sine values is essential for analyzing waves, circular motion, and even in sound and light studies. For accuracy in your results, always round to the required decimal places as instructed.

Angle Measurement

Understanding how angles are measured is crucial for solving many mathematical problems. Angles can be measured in degrees or radians, each having its applications.
Degrees are more intuitive and commonly used in daily life. A full circle is 360 degrees, with each degree further divisible into 60 minutes, and each minute into 60 seconds.

• Great for simple geometry problems.
• Used mainly in schools and certain applications like navigation.

In contrast, radians are more prevalent in advanced math and sciences. Instead of breaking things into degrees, radians consider the radius of a circle.

• For radians, a full circle is \( 2\pi \), making \( \pi \) radians equal to 180 degrees.
• Radians provide a direct relationship between the angle and the arc length, making them ideal in calculus and theoretical physics.

Choosing between degrees and radians depends on the context of your problem, the tools at your disposal, and what makes interpreting results easier.