Problem 21 - Algebra 2 and Trigonometry

Question

In \(3-38,\) find each function value to four decimal places. $$ \cos 18^{\circ} 12^{\prime} $$

Step-by-Step Solution

Answer

\(\cos 18^{\circ} 12^{\prime} \approx 0.9500\)

Step 1: Convert Minutes to Decimal Degrees

We are given an angle of \(18^{\circ} 12^{\prime}\). The goal is to find \(\cos 18^{\circ} 12^{\prime}\). First, we need to convert the angle into decimal degrees. Since there are 60 minutes in a degree, we convert \(12^{\prime}\) to decimal form by using \(12^{\prime} = \frac{12}{60} = 0.2^{\circ}\).

Step 2: Combine Into One Degree Measure

Now, we need to add the degrees together. The original degrees \(18^{\circ}\) plus the converted minutes \(0.2^{\circ}\) results in \(18.2^{\circ}\). So, \(18^{\circ} 12^{\prime} = 18.2^{\circ}\).

Step 3: Calculate the Cosine

Now, use a calculator to find the cosine. Input the decimal degrees into the calculator to get \(\cos 18.2^{\circ}\). On a scientific calculator, you typically ensure you are in degree mode and then calculate: \(\cos 18.2^{\circ} \approx 0.9500\).

Step 4: Round the Result

Finally, round the result to four decimal places. In this case, \(0.9500\) is already in four decimal places.

Key Concepts

Angle Conversion

When working with angles, it is crucial to convert them accurately between different units. Typically, angles can be expressed in degrees and minutes, where 1 degree is equivalent to 60 minutes. For calculations like finding the cosine, it is often necessary to convert the angle into "decimal degrees," which is a straightforward decimal number.

• To convert minutes to decimal degrees, divide the number of minutes by 60, since there are 60 minutes in one full degree. For instance, if you have 12 minutes, it would be converted as \( \frac{12}{60} = 0.2 \degree \).
• Combine this decimal with the whole degree value. For example, \(18^{\circ} 12^{\prime} \) becomes \( 18^{\circ} + 0.2^{\circ} = 18.2^{\circ} \).

Remember this simple conversion method, as it is a common task in many mathematical and scientific situations involving angles.

Decimal Degrees

Decimal degrees simplify the representation of angles and make calculations easier, especially when using calculators or computer software. Unlike the traditional degrees, minutes, and seconds (DMS) format, decimal degrees offer a single continuous number.
Here’s what makes decimal degrees beneficial:

• They eliminate the need for handling three separate components: degrees, minutes, and seconds. You have just one value like 18.2°.
• They are easier to use in arithmetic operations since everything is in the same base (base 10).
• Software and calculators often prefer decimal degrees for trigonometric calculations, which is why it's essential to understand how to convert your values to this format.

Decimal degrees ultimately make your mathematical life easier, saving time and minimizing errors.

Scientific Calculator

Using a scientific calculator correctly is key when working with mathematical problems, such as finding trigonometric functions of angles. First, ensure your calculator is set to "degree mode," especially when feeding it with angles not in radians.

• To switch modes, look for a button on your calculator display labeled as DEG (Degree), RAD (Radians), or possibly GRAD (Gradians). Ensure it reads "DEG."
• Input the angle in decimal degrees - for instance, type 18.2 if your angle is 18.2 degrees.
• Press the cosine button, often labeled as "COS," to perform the cosine function. The calculator then processes this information in degree mode to provide a result like 0.9500.

Knowing these steps will help you efficiently utilize your scientific calculator for various trigonometric evaluations.

Trigonometric Function Evaluation

Evaluating trigonometric functions like cosine involves understanding both the function and its input - the angle, often in degrees. The cosine function specifically relates to the x-coordinate of a point on the unit circle corresponding to the angle measured from the positive x-axis.
Steps to evaluate cosine using a decimal-degree angle:

• Convert your angle into decimal degrees, if it's not already. For example, \(18^{\circ} 12^{\prime}\) converts to \(18.2^{\circ}\).
• Use a calculator, in degree mode, to find the trigonometric value. In this scenario, enter the value 18.2 and use the cosine (COS) button.
• Read and round your calculated result to the needed decimal places. For accuracy, processes often require rounding to four decimal places. Here, \( cos 18.2^{\circ} \approx 0.9500 \).

Understanding these fundamental steps aids in evaluating cosine and other trigonometric functions swiftly and precisely.