Problem 74 - Algebra 2

Question

Write each measure in radians. Express the answer in terms of \(\pi\) and as a decimal rounded to the nearest hundredth. $$ 200^{\circ} $$

Step-by-Step Solution

Answer

In terms of \(\pi\), an angle of 200° is \((10/9)*\pi\) radians. As a decimal rounded to the nearest hundredth place, this is approximately 3.49 radians.

Step 1: Convert the angle measurement from degrees to radians

To convert from degrees to radians, we use the conversion factor \(\pi/180\). So, for an angle of 200 degrees, the conversion will be: 200° * \(\pi/180\).

Step 2: Simplify the expression.

Perform the multiplication: \(200*\pi / 180 = \((20/18) * \pi\) = \((10/9) * \pi\)\). Here, the angle is expressed as a multiple of \(\pi\). This provides the exact value.

Step 3: Decimal approximation up to two decimal places.

We calculate the decimal approximation by substituting \(\pi\) with approximately 3.14. Therefore, \((10/9) * \pi\) approximately equals 3.49 (to two decimal places). Thus, there is a rounded representation of the original angle measure in radians.

Key Concepts

Degrees to Radians

When converting angles from degrees to radians, it's essential to use the correct conversion factor. The key to understanding this process is recognizing that a full circle has 360 degrees, which is equivalent to \(2\pi\) radians. Thus, to convert degrees into radians, we utilize the relation:

• \(1^{\circ} = \frac{\pi}{180}\) radians

Given an angle in degrees, the formula to convert it into radians becomes:

• \(\text{radians} = \text{degrees} \times \frac{\pi}{180}\)

This formula helps to determine how many "\(\pi\) sections" make up the measured angle. For example, with an angle of \(200^{\circ}\), multiplying \(200\) by \(\frac{\pi}{180}\) results in \(\frac{200\pi}{180}\). This allows us to express angles in a standard unit regardless of the distance or size of the circle.

Pi in Calculations

The value of \(\pi\) plays an enormous role in trigonometry and geometry, particularly when converting angles. In calculations, \(\pi\) is utilized as the ratio of a circle's circumference to its diameter. It is commonly approximated as 3.14, but it is an irrational number, meaning it cannot be fully represented as a finite decimal or fraction.

When we express angles in radians, \(\pi\) becomes part of the angle measurement. For instance, when converting \(200^{\circ}\) to radians, you might end up with \((10/9)\pi\). This is a cleaner way of expressing the angle that avoids approximation in this intermediate step. It allows for precise angle communication between varying mathematical and scientific calculations.

• Use \(\pi = 3.14159...\) or the symbol \(\pi\) itself for exactness.
• Replace \(\pi\) with a simplified number only when a decimal approximation is necessary.

Decimal Approximation

Once you have the value in terms of \(\pi\), sometimes real-world applications need a decimal approximation. Let's take the calculation from before: \((10/9)\pi\). To find a decimal approximation, use \(\pi \approx 3.14159\):

• First, find the product: \((10/9) \times 3.14159\) to get approximately 3.49065.
• Then, round this to the nearest hundredth to find 3.49.

Converting the radian measure to a decimal makes it easier to gauge the angle's size in practical terms. Decimal approximations, while not exact, are often "good enough" for engineering, navigation, and real-world geometry where high precision is unnecessary.
Remember:

• Decimal values are approximations and should be clearly marked as rounded or approximate.
• Always state the degree of precision, here to the nearest hundredth.