Problem 63 - Algebra and Trigonometry Real Mathematics, Real People

Question

Rewrite each angle in radian measure as a multiple of \(\pi .\) (Do not use a calculator.) (a) \(30^{\circ}\) (b) \(150^{\circ}\)

Step-by-Step Solution

Answer

(a) \(\frac{\pi}{6}\) (b) \(\frac{5\pi}{6}\)

Step 1: Conversion of \(30^{\circ}\) to radian measure

To convert degrees to radians, you multiply the degree value by \(\frac{\pi}{180}\). \[30^{\circ} * \frac{\pi}{180} = \frac{\pi}{6}\]

Step 2: Conversion of \(150^{\circ}\) to radian measure

Again, to convert degrees to radians, multiply the degree value by \(\frac{\pi}{180}\). \[150^{\circ} * \frac{\pi}{180} = \frac{5\pi}{6}\]

Key Concepts

Degree to Radian Conversion

Converting between degrees and radians is essential for accurate angle measurement in trigonometry. The conversion is quite straightforward. You multiply the degree by \( \frac{\pi}{180} \) to convert it into radians. This formula is derived from the fact that a full circle in degrees is 360° and in radians, it is \(2\pi\). Therefore:

• A half-circle is 180° which is equivalent to \(\pi\) in radians.
• So when you want to find what one degree is in radians, you divide \(\pi\) by 180. This gives \(\frac{\pi}{180}\).
• To convert back, you'd use the reciprocal: multiply radians by \(\frac{180}{\pi}\) to get degrees.

This method helps in finding exact radian measures without a calculator, making it more efficient when dealing with common angles.

Angle Measurement

Angle measurement is the process of measuring the size of an angle. Angles can be measured in either degrees or radians, which are the most common units. Understanding these units is crucial in fields like geometry and physics.

• Degrees: This is a more ancient unit where a full circle is divided into 360 parts, each part being one degree.
• Radians: Involves the arc length of a circle's radius. A circle has \(2\pi\) radians.

Radians often make it easier to perform mathematical operations, especially in calculus and trigonometry, because they relate more directly to real numbers and mathematical constants.
Both degrees and radians serve their purpose depending on the application and context. For example, degrees are often used in navigation and everyday contexts, while radians are preferred in advanced mathematics and scientific calculations.

Pi

Pi, represented by the symbol \(\pi\), is a mathematical constant that describes the ratio of a circle’s circumference to its diameter. It is an irrational number, which means it has an infinite number of digits beyond the decimal point without repeating. The value is approximately 3.14159.
Pi is a central component in various mathematical formulas—especially those involving circles and periodic functions like sine and cosine.

• Circumference: The formula for finding the circumference of a circle is \(C = 2\pi r\), where \(r\) is the radius.
• Area: The area of a circle can be found using \(A = \pi r^2\).
• Trigonometry: Functions like sine, cosine, and tangent relate to pi when analyzing angles and oscillations.

In the context of radians, pi helps define one complete rotation as \(2\pi\), simplifying mathematical notation in the study of angles and circular motion.