Problem 26 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

\(\cos(205^{\circ} 12^{\prime}) \approx -0.4702\)

Step 1: Convert degrees and minutes to decimal degrees

The angle is given as \(205^{\circ} 12^{\prime}\). To convert it to decimal degrees, divide the minutes by 60 and add to the degrees. So, \(205^{\circ} 12^{\prime} = 205 + \frac{12}{60} = 205.2^{\circ}\).

Step 2: Use cosine function

To find \(\cos(205.2^{\circ})\), use a calculator. Enter \(205.2\) into the cosine function of your calculator.

Step 3: Calculate and round the result

Calculating \(\cos(205.2^{\circ})\) gives a result of approximately -0.4702. Ensure the calculator is in degree mode and round the answer to four decimal places.

Key Concepts

Degree to Decimal Conversion

When it comes to trigonometric functions, often we are given angles in a format that combines degrees and minutes, like in this problem. Converting these values from degree-minute form into a straightforward decimal degree form is essential for calculations.
To convert an angle given in degrees and minutes into just degrees:

• Keep the degree portion as it is.
• Divide the number of minutes by 60 since there are 60 minutes in a degree.
• Add the result to the degree portion to get the decimal form of the angle.

Here's how it works in the given problem: From the angle of \( 205^{\circ} 12^{\prime} \), divide the 12 minutes by 60, resulting in \( 0.2 \) degrees. Adding it to the original 205 degrees gives \( 205.2^{\circ} \).
This decimal representation is what you need to use when entering angles into a calculator for trigonometric functions.

Cosine Function

The cosine function is one of the basic trigonometric functions. It is often used to find the cosine of an angle, which represents the ratio of the adjacent side to the hypotenuse in a right triangle.
This function is vital in calculating various properties in geometry, physics, engineering, and more. When you input an angle into the cosine function of a calculator:

• Ensure that the calculator is set to the right mode depending on whether you're using degrees or radians (in our case, it should be in degree mode).
• Input the decimal degree value into the calculator.
• Simply press the cosine function button to get the result.

For our specific exercise, after converting \( 205^{\circ} 12^{\prime} \) to \( 205.2^{\circ} \), input this value into the cosine function. The calculator will return the cosine value numerically.

Angle Measurement

Understanding angle measurements is paramount when dealing with trigonometry. Angles can be measured in various units including degrees, radians, and sometimes gradians.
Degrees, as used here, are further expressed in a sexagesimal system involving degrees, minutes, and sometimes seconds. This highlights:

• The full circle is divided into 360 parts, each being a degree.
• Each degree is split into 60 minutes.
• This hierarchical structure allows precise measurement for complex applications.

In trigonometric calculations, knowing how to express these angles properly is crucial as it directly affects the outcomes of calculations.
Ensuring correct conversion and calculator settings for these units will help you arrive at the right calculations, as shown in this exercise's conversion of \( 205^{\circ} 12^{\prime} \) to its decimal form for further computation.