Problem 71 - Algebra 2

Question

Write each measure in radians. Express the answer in terms of \(\pi\) and as a decimal rounded to the nearest hundredth. $$ -15^{\circ} $$

Step-by-Step Solution

Answer

The measure of -15 degrees in radians is \(-\frac{1}{12}\pi\) or approximately -0.26

Step 1: Convert to Radians in Terms of \(\pi\)

Use the conversion factor that 180 degrees is equivalent to \(\pi\) radians. That is, divide -15 by 180 and multiply by \(\pi\). So, \[-15^{\circ} = -15 \times \frac{\pi}{180} = -\frac{1}{12}\pi\] radians.

Step 2: Convert to Decimal Form

Substitute \(\pi\) with its approximate decimal value 3.14 and round the answer to nearest hundredth. So, \[-\frac{1}{12}\pi = -\frac{1}{12}\times 3.14 \approx -0.26\] radians.

Key Concepts

Radians

Radians are a way to measure angles, similar to degrees. While degrees are the traditional unit of measuring angles, radians offer a different perspective. A circle measures 360 degrees, but in radians, a full circle is divided into approximately 6.283 radians, which is equivalent to \(2\pi\) radians. This relationship is important because it links the circle's radius directly to the circumference, making calculations in trigonometry simpler. In essence, one radian is the angle made at the center of a circle by an arc whose length is equal to the circle's radius. Thus, this unit is deeply tied to the geometry of circles.

Degrees to Radians Conversion

Converting degrees to radians is straightforward, thanks to the constant relationship between the two. We know that 180 degrees equal \(\pi\) radians. This means any angle in degrees can be converted to radians by multiplying by \(\frac{\pi}{180}\). For example, to convert \(-15^{\circ}\) to radians:

• First, divide \(-15\) by 180, which gives \(-\frac{1}{12}\).
• Then, multiply by \(\pi\), resulting in \(-\frac{1}{12}\pi\) radians.

This conversion is crucial for solving many mathematical problems, especially in calculus and trigonometry, where radians are often preferred.

Decimal Approximation of Pi

Pi (\(\pi\)) is a mathematical constant that represents the ratio of a circle's circumference to its diameter. Its value is approximately 3.14159, but for simpler calculations, it is common to use a rounded version, such as 3.14. This approximation is handy when converting angles to decimal form or when precision isn’t critical.To express angles in radians as decimals, we replace \(\pi\) with its approximate value. In our earlier example of \(-\frac{1}{12}\pi\) radians:

• First, substitute \(\pi\) with 3.14.
• Calculate \(-\frac{1}{12}\times 3.14\) to get about \(-0.26\) radians.

This process shows how an angle can be represented both as a multiple of \(\pi\) and as a decimal, offering flexibility in different mathematical contexts.