Problem 27 - Algebra 2 and Trigonometry

Question

In \(3-38,\) find each function value to four decimal places. $$ \sin 205^{\circ} 12^{\prime} $$

Step-by-Step Solution

Answer

\( \sin(205^{\circ}12^{\prime}) \approx -0.9063 \)

Step 1: Convert Degrees and Minutes to Decimal Degrees

The given angle is in degrees and minutes. First, convert the angle into decimal degrees: \[ 205^{\circ}12^{\prime} = 205^{\circ} + \frac{12}{60}^{\circ} = 205.2^{\circ} \]

Step 2: Use the Sine Function

Use the sine function on a calculator or appropriate tool to find \( \sin(205.2^{\circ}) \). Ensure your calculator is set to degree mode.

Step 3: Calculate the Sine Value

Calculate the value using a calculator:\[\sin(205.2^{\circ}) \approx -0.9063\] Round the result to four decimal places, which is already done here.

Key Concepts

Sine Function

The sine function is a fundamental component of trigonometry and deals with the relationship between angles and sides in right-angled triangles. Essentially, for a given angle, it expresses the ratio of the opposite side to the hypotenuse. The sine function can be written as \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \), where \( \theta \) represents the angle in question.

• The sine function is periodic, meaning its values repeat at regular intervals. This interval is typically \( 360^{\circ} \) or \( 2\pi \) radians.
• The output range of the sine function is between \(-1\) and \(1\), capturing the full spectrum of vertical angular movements.
• The sine function is useful not only in theoretical math but also in practical applications like engineering and physics, particularly when dealing with waveforms and oscillations.

The exercise example uses the sine function to find the value of an angle expressed in degrees and minutes. Being familiar with this function helps solve problems related to periodic phenomena and circular motion.

Angle Conversion

Angles can be represented in various forms, such as degrees, radians, or even as degrees and minutes. In the given exercise, the angle is initially expressed in degrees and minutes. Here's what you need to know about converting between these formats:To convert an angle from degrees and minutes to decimal degrees, follow these steps:

• Note that one full degree equals \( 60 \) minutes (just as an hour equals \( 60 \) minutes).
• Divide the number of minutes by \( 60 \) to convert them into a fraction of a degree.
• Add this fraction to the degrees to get the angle in decimal degrees.

For instance, the angle \( 205^{\circ}12^{\prime} \) can be calculated as: \[ 205^{\circ}12^{\prime} = 205^{\circ} + \frac{12}{60}^{\circ} = 205.2^{\circ} \]. This conversion ensures you have a consistent format for further calculations.Understanding angle conversion is critical in achieving precise calculations and is often required when using certain mathematical functions that operate on angles. This step ensures consistency and accuracy, especially when inputs are needed for functions like sine, cosine, or tangent.

Degree Mode

When performing calculations involving angles, it is crucial to ensure that your calculator is set to the correct mode: degree mode or radian mode. This setting determines how the calculator interprets angle measurements. For trigonometric calculations involving angles given in degrees, such as in the exercise under discussion:

• Ensure that the calculator or software tool is set to "degree mode" so that the inputs are processed correctly.
• Degree mode interprets angles in terms of degrees (as opposed to radians or other measurements).
• Without setting the calculator to the proper mode, the output won't accurately reflect the given angle's trigonometric function.

Most calculators have a button or a setting in the menu where you can toggle between degree and radian modes. This step is fundamental but often overlooked, leading to errors in calculations, particularly in settings like exams and important assignments.So remember, for angles in degrees, it’s degree mode to get the correct trigonometric values. This will help you achieve the most accurate results with your calculations, as demonstrated in the exercise solution where the sine of \( 205.2^{\circ} \) was found.