Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-38,\) find each function value to four decimal places. $$ \cos 100^{\circ} $$

Step-by-Step Solution

Answer

The value of \(\cos 100^{\circ}\) to four decimal places is \(-0.1736\).

Step 1: Understanding the Problem

We need to find the cosine of an angle given in degrees, specifically \(100^{\circ}\). We will use a scientific calculator to compute this value accurately to four decimal places.

Step 2: Set Calculator to Degree Mode

Before performing the calculation, ensure your calculator is set to degree mode because the angle is given in degrees. This is crucial for obtaining the correct trigonometric value.

Step 3: Calculate Cosine Value

Using a calculator, enter \(100\) and then press the cosine button. The calculator should display the cosine value of \(100^{\circ}\).

Step 4: Round the Result

The calculator will give a long decimal, but we only need four decimal places. If necessary, round the fifth digit appropriately to get an accurate four decimal place result.

Key Concepts

Understanding Degree Mode

When calculating trigonometric functions such as cosine, it's crucial to ensure your calculator is set to "degree mode." This mode is essential when your angle measurements are provided in degrees, as opposed to radians. Degrees and radians are two different units for measuring angles.

• Degrees are based on dividing a circle into 360 equal parts.
• Radians are based on the radius of a circle. A full circle is equivalent to \(2\pi\) radians.

Using the wrong mode could lead to incorrect results because the basis for the angle measurement would be misinterpreted. On most calculators, you can typically switch to degree mode by pressing a button labeled "Deg" or ensuring a "D" appears on the display screen.
This ensures that when you input an angle of \(100^{\circ}\), the calculator understands it correctly and provides the precise trigonometric values.

Calculating Trigonometric Values

Trigonometric functions such as sine, cosine, and tangent are fundamental in mathematics for dealing with angles and relationships in right triangles. The cosine function in particular measures the horizontal distance from the origin to a point on the unit circle that corresponds to a given angle.
For an angle such as \(100^{\circ}\), the cosine value tells us how far 'right' or 'left' the terminal segment of the angle is extended on the unit circle.

• If a calculator is correctly set to degree mode, it computes this value by determining the coordinate of the intersection point on the unit circle for that angle.
• Once inputting \(100\) and pressing the cosine button, the calculator will display a value which often needs to be rounded.

This process gives the cosine of \(100^{\circ}\) around \(-0.1736\), before rounding to the fourth decimal place.

Rounding Decimals Correctly

After obtaining a numerical output from your calculator, the task is to present the result accurately to four decimal places. This step is crucial for precise mathematical communication and comparison.
Rounding involves adjusting a number to make it simpler, but still close in value to the original number. When rounding:

• Identify the fifth decimal place.
• If this digit is 5 or greater, increase the fourth decimal place by one.
• If it is less than 5, leave the fourth decimal place as is.

For example, if the calculator displays \(-0.173648\), you will round it to \(-0.1736\) because the fifth decimal is 4.
Precision can be vital, especially in scientific contexts where slight differences can alter outcomes significantly.