Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-38,\) find each function value to four decimal places. $$ \sin 28^{\circ} $$

Step-by-Step Solution

Answer

The sine of 28 degrees is approximately 0.4695.

Step 1: Understand the Problem

We are asked to find the sine value of an angle, specifically the sine of 28 degrees. This involves using the sine function, typically accessed via a scientific calculator or trigonometric tables.

Step 2: Convert Angle to Radians (if needed)

The sine function can be applied directly to degrees in most calculators, so conversion to radians is not necessary for this exercise. We proceed with the degree measure.

Step 3: Use a Calculator

Using a scientific calculator, enter the angle 28 degrees and apply the sine function. This means pressing the 'sin' button after ensuring the calculator is in degree mode.

Step 4: Compute the Sine Value

1. Make sure the calculator is set to "Degree" mode. 2. Enter 28 and press the 'sin' button. 3. The calculator will output the sine value of 28 degrees.

Step 5: Round the Result

After obtaining the sine value from the calculator, round it to four decimal places as instructed. This ensures precision as per the exercise requirement.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions. It helps us understand the relationship between angles and the ratios of sides in a right triangle. When you use the sine function, you're essentially looking to find the length of the side opposite the angle relative to the hypotenuse of the triangle. This relationship is expressed in the formula:\[ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \]Here, \(\theta\) is the angle in question.

• In our exercise, \(\theta\) is 28 degrees.
• The process involves using this angle to compute the side's ratio.

While sine is visually and practically understood through triangles, its applications stretch beyond just geometry. It's widely used in fields such as physics, engineering, and even computer science for modeling wave patterns and more. So, mastering the sine function can open a wide range of practical applications.

Degree Mode

For accurate trigonometric calculations, it's crucial that your calculator is set to the correct mode, particularly 'Degree Mode' when dealing with angles measured in degrees. Most scientific calculators allow calculations in three modes: degrees, radians, and sometimes gradients. Using 'Degree Mode' means the calculator interprets angles as degrees.

• Always double-check your calculator before calculation to avoid errors.
• Many calculators have a 'MODE' button that allows you to toggle between different angle units.
• There is usually some indication on the display showing which mode you're currently in, often a letter such as 'D' for degrees.

Ensuring the correct mode setting saves time and prevents calculation errors that can stem from treating degrees as another unit, such as radians. Getting comfortable with your calculator's functions will make solving trigonometry problems much smoother.

Rounding Decimal Places

Rounding decimal places is a common practice in mathematics for expressing results cleanly and precisely. In our exercise, we are required to round the sine value to four decimal places. Here's how you do it:

• First, compute the value using your calculator.
• Look at the fifth decimal place to decide whether to round up or leave it as is.
• If the fifth digit is 5 or higher, increase the fourth decimal place by one.
• For example, if the sine of 28 degrees is calculated as 0.469471, you round it to 0.4695.

Rounding ensures that your final result is both easier to handle and consistent with precise requirements. Practicing rounding to different decimal places can help you become adept at managing significant figures, which is key in controlled scientific calculations.