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Magazine Chapter 8: Problem 80
Convert to radian measure. $$-225^{\circ}$$
Short Answer
Expert verified
-袄蹿谤补肠调5蟺皑调4皑
Step by step solution
01
- Understanding Degree to Radian Conversion
To convert degrees to radians, use the conversion factor: \( \frac{\text{蟺}}{180^{\text{掳}}} \). This factor arises because \( 180^{\text{掳}} \) is equal to 蟺 radians.
02
- Apply the Conversion Factor
Multiply the degree measure by \( \frac{\text{蟺}}{180^{\text{掳}}} \): \( -225^{\text{掳}} \times \frac{\text{蟺}}{180^{\text{掳}}} \).
03
- Simplify the Fraction
Simplify the fraction formed by the multiplication: \( \frac{-225蟺}{180} \). This can be simplified by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 45.
04
- Perform the Division
Divide both the numerator and the denominator by 45: \( \frac{-225蟺}{180} = \frac{-5蟺}{4} \).
05
- Write the Final Answer
The radian measure for \( -225^{\text{掳}} \) is \( -袄蹿谤补肠调5蟺皑调4皑 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Radian Measure
Radian measure is a way of expressing angles using the radius of a circle. One radian is the angle formed when the radius is wrapped around the circle's circumference. Unlike degrees, which divide a circle into 360 equal parts, radians measure angles based on the actual distance traveled along the circle's edge. Here's a simple way to think about it: if you stretch the radius length along the circle's boundary, the angle created at the center is one radian. Radians provide a more natural framework for mathematics, especially in trigonometry and calculus. In fact, one complete revolution around a circle is equal to \( 2蟺 \) radians.
Degree Measure
Degree measure is a traditional way to describe angles by dividing a circle into 360 equal parts. Each part is one degree (\(1^{\text{掳}}\)). This system is believed to have been derived from ancient Babylonians, who used a base-60 number system. When dealing with degrees, it鈥檚 simple to visualize angles: \(-90^{\text{掳}}\) is a quarter circle, \(180^{\text{掳}}\) is a half circle, and \(-225^{\text{掳}}\) extends below the negative y-axis by half a quarter circle more. While degrees are common in navigation and daily activities, converting them into radians can simplify calculations, especially when dealing with trigonometric functions.
Pi (蟺)
Pi (蟺) is a special mathematical constant and represents the ratio of a circle's circumference to its diameter. It's an irrational number, meaning it cannot be expressed as a simple fraction and its decimal form goes on forever without repeating. To approximate, we often use \(\text{蟺} \approx 3.14159\). Pi is fundamental in mathematics, appearing in many formulas involving circles. For angle conversions, \(蟺\) radians correspond to \(180^{\text{掳}}\). That鈥檚 why the conversion factor \(\frac{蟺}{180^{\text{掳}}} \) is so crucial for translating degrees to radians seamlessly.
Greatest Common Divisor (GCD)
The Greatest Common Divisor (GCD) of two numbers is the largest positive integer that divides both numbers without leaving a remainder. It鈥檚 a key concept in simplifying fractions. For instance, the fraction \(\frac{-225蟺}{180} \) can be simplified by determining the GCD of \(-225 \) and \(180\), which is \(45\). Dividing both the numerator and the denominator by 45 yields the simplified fraction \(\frac{-5蟺}{4}\). Simplification is crucial because it makes understanding and working with numbers easier, reducing complexity in both mathematical operations and their practical applications.
