Problem 70 - Algebra 2

Question

Write each measure in radians. Express the answer in terms of \(\pi\) and as a decimal rounded to the nearest hundredth. $$ -50^{\circ} $$

Step-by-Step Solution

Answer

Therefore, \(-50^{\circ}\) is equivalent to \(\frac{-5\pi}{18}\) radians or approximately \(-0.87\) radians.

Step 1: Identify what is given

The problem has already provided an angle in degrees, which is -50 degrees.

Step 2: Utilize the conversion factor

To convert from degrees to radians, multiply the degrees by \(\frac{\pi}{180}\). So the calculation is \((-50) \cdot \frac{\pi}{180}\), which simplifies to \(\frac{-5\pi}{18}\). Therefore, the expression of -50 degrees in terms of \(\pi\) is \(\frac{-5\pi}{18}\).

Step 3: Convert the answer to decimal

To convert this into decimal form, substitute \(\pi\) with approximately 3.14, so \(\frac{-5\pi}{18}\) becomes approximately \((-5) * 3.14/18 = -0.87\).

Key Concepts

Degrees to Radians

When trying to convert an angle from degrees to radians, it is essential to understand the relationship between the two units.
Angles can be measured in degrees or radians, and they represent different systems of measuring angle size.

One full rotation around a circle is 360 degrees or 2\(\pi\) radians. Therefore, these measurement systems are related through the conversion factor \(\frac{\pi}{180}\).

• This factor helps convert degrees to radians by multiplying the degree measure by \(\frac{\pi}{180}\).
• This reflects the ratio between the full circle in degrees (360) and in radians (2\(\pi\)).

Using this conversion factor, you can easily change any degree measure to radians. Just remember: multiply your degree measure by \(\frac{\pi}{180}\), and you will express it in radians.

Pi

Pi (commonly known by its symbol \(\pi\)) is an essential constant in mathematics and is approximately equal to 3.14159.

Pi is an irrational number, which means it cannot be expressed as a simple fraction, and its decimal representation is non-repeating and non-terminating.

• It is commonly used in calculations involving circles since \(\pi\) represents the ratio of a circle's circumference to its diameter.
• This makes \(\pi\) central to trigonometry and the conversion between degrees and radians.

In problems related to angle conversion like our exercise, \(\pi\) allows the degree measures to be expressed in terms of radians, maintaining a precise mathematical expression.
When solving these problems, you keep \(\pi\) intact in the expression to retain the exactness (like \(\frac{-5\pi}{18}\)).
Then, for practical purposes or approximation, you can substitute it with 3.14 or a more accurate value of \(\pi\) to find a decimal approximation.

Decimal Approximation

Decimal approximation is the process of estimating the value of a number, particularly irrational numbers like \(\pi\), in decimal form up to a desired precision.
This is useful in providing a more intuitive and practical understanding of mathematical expressions.

For example, in our exercise, once you have the angle in radians \(\frac{-5\pi}{18}\), you might want to express it as a decimal to better grasp its magnitude.

• Substitute \(\pi\) with 3.14 to get an approximate decimal form.
• Perform the multiplication and division: \((-5) \cdot 3.14/18\).

In this example, the approximate decimal result is \(-0.87\), rounded to the nearest hundredth.
Using decimal approximations helps when you require a quick or practical estimate instead of an exact formula, making it helpful in everyday calculations.