Problem 15 - Precalculus Mathematics for Calculus

Question

Find the degree measure of the angle with the given radian measure. $$\frac{7 \pi}{6}$$

Step-by-Step Solution

Answer

210 degrees

Step 1: Understanding Radian to Degree Conversion

To convert an angle from radians to degrees, we use the conversion factor \( \frac{180}{\pi} \). This works because \( \pi \, ext{radians} = 180^\circ \). So, any radian measure can be converted to degrees by multiplying by this factor.

Step 2: Set Up the Conversion Equation

Start with the given radian measure \( \frac{7 \pi}{6} \) and set up the equation to convert it to degrees: \( \text{angle in degrees} = \frac{7 \pi}{6} \times \frac{180}{\pi} \).

Step 3: Simplify the Expression

Cancel out \( \pi \) from the numerator and denominator: \( \frac{7 \pi}{6} \times \frac{180}{\pi} = \frac{7}{6} \times 180 \).

Step 4: Perform the Multiplication

Multiply 7 by 180: \( 7 \times 180 = 1260 \). Divide the result by 6: \( \frac{1260}{6} = 210 \).

Step 5: Conclusion

Therefore, the angle \( \frac{7 \pi}{6} \) radians is equivalent to 210 degrees.

Key Concepts

Angle Measurement

Understanding how we measure angles is crucial for solving problems in mathematics, especially in trigonometry and geometry. There are two main units of angle measurement: degrees and radians. A circle is divided into 360 degrees, which means a full rotation around any circle is a complete 360-degree turn.
Radians, on the other hand, are based on the radius of the circle. When the arc length is equal to the radius, the angle is one radian. Thus, in one full circle, there are \(2\pi\) radians, which equates to the 360 degrees a circle encompasses. This relationship is critical for converting between these two units.

• A right angle is 90 degrees or \(\frac{\pi}{2}\) radians.
• A straight angle is 180 degrees or \(\pi\) radians.
• A full circle measures 360 degrees or \(2\pi\) radians.

Knowing these basics allows you to quickly estimate angles in either degrees or radians.

Trigonometry

Trigonometry is a branch of mathematics that explores the relationships between angles and distances. It is essential in fields ranging from physics to engineering. The subject focuses on the properties of triangles, specifically right-angled triangles. It deals with ratios of sides rather than the sides themselves.
In any right-angle triangle, there are three primary trigonometric functions:

• Sine (\(\sin\)) is the opposite side over the hypotenuse.
• Cosine (\(\cos\)) is the adjacent side over the hypotenuse.
• Tangent (\(\tan\)) is the opposite side over the adjacent side.

Angles in trigonometry are usually measured in radians, which simplifies many mathematical applications. Understanding how to convert radians into degrees and vice versa helps in solving complex trigonometric problems efficiently. This makes learning the conversion process all the more valuable for students.

Mathematical Conversion

Converting between different units is a fundamental skill in mathematics. When it comes to angles, converting from radians to degrees, or vice versa, is very common. To convert a radian measure into degrees, you use the conversion factor \(\frac{180}{\pi}\). This factor originates from the relationship that \(\pi\text{ radians} = 180^\circ\).
Here's how you do a conversion:

• Start with the radian measure. For instance, consider \(\frac{7\pi}{6}\).
• Multiply this by the conversion factor: \(\frac{7\pi}{6} \times \frac{180}{\pi}\).
• Simplify the equation by canceling out \(\pi\): \(\frac{7}{6} \times 180\).
• Perform the multiplication to find the equivalent degree measure: \(1260/6 = 210^\circ\).

Mastering this process allows you to fluently switch between different units, making computations easier and interpretations clearer.