Problem 11 - Precalculus: Functions and Graphs

Question

Exer. \(9-12:\) Find the exact radian measure of the angle. (a) \(450^{\circ}\) (b) \(72^{\circ}\) (c) \(100^{\circ}\)

Step-by-Step Solution

Answer

(a) \(\frac{5\pi}{2}\), (b) \(\frac{2\pi}{5}\), (c) \(\frac{5\pi}{9}\).

Step 1: Understanding Degree to Radian Conversion

To convert an angle from degrees to radians, we use the formula \( \text{radians} = \left( \text{degrees} \times \frac{\pi}{180} \right) \). This formula is derived from the fact that \(180^{\circ}\) is equal to \( \pi \) radians.

Step 2: Convert 450° to Radians

Using the conversion formula, \( 450^{\circ} \times \frac{\pi}{180} \). Simplifying, we have \( \frac{450}{180} \times \pi = \frac{5\pi}{2} \). So, the angle in radians is \( \frac{5\pi}{2} \).

Step 3: Convert 72° to Radians

Applying the conversion formula, \( 72^{\circ} \times \frac{\pi}{180} \). Simplifying, we get \( \frac{72}{180} \times \pi = \frac{2\pi}{5} \). Therefore, the angle in radians is \( \frac{2\pi}{5} \).

Step 4: Convert 100° to Radians

Using the same conversion formula, \( 100^{\circ} \times \frac{\pi}{180} \). Simplify to obtain \( \frac{100}{180} \times \pi = \frac{5\pi}{9} \). Hence, the angle in radians is \( \frac{5\pi}{9} \).

Key Concepts

Radian Measure

The concept of radian measure is a fundamental aspect of trigonometry and angle measurement. Radians provide a way to express angles based on the radius of a circle. Instead of using degrees, which is a more arbitrary unit, radians relate directly to the geometry of the circle.
One radian is the angle created when the arc length is equal to the radius of the circle. Since the circumference of a circle is given by the formula \(2\pi r\), there are \(2\pi\) radians in a full circle or \(360^{\circ}\).

• In terms of fraction conversion, \(180^{\circ} = \pi\) radians.
• This relationship helps relate degree measures to radian measures easily with simple multiplication.

Understanding radians is crucial for trigonometry, as they allow for more straightforward and natural manipulation when dealing with periodic functions and calculus.

Angle Conversion

Angle conversion between degrees and radians is essential in trigonometry and other mathematical disciplines. To convert degrees to radians, use the formula:
\[\text{radians} = \left( \text{degrees} \times \frac{\pi}{180} \right)\]
This formula arises because \(180^{\circ}\) is equivalent to \(\pi\) radians. Below are some key points about conversion:

• Multiply the degree measure by \(\frac{\pi}{180}\) to get radians.
• To convert back from radians to degrees, multiply the radians by \(\frac{180}{\pi}\).

For example, to convert \(450^{\circ}\) to radians, calculate:
\[450^{\circ} \times \frac{\pi}{180} = \frac{5\pi}{2}\]
This shows how an angle greater than \(360^{\circ}\) (a full circle) can be easily translated into radian measure.

Trigonometry

Trigonometry links mathematics with angles, circles, and periodic functions. It plays a crucial role in understanding wave forms, oscillations, and rotations. The radian system seamlessly integrates with trigonometry, providing elegant solutions to many problems.
Some essential trigonometric principles involve sine, cosine, and tangent functions:

• These functions often operate most naturally using radians rather than degrees.
• With radians, periodic functions become more intuitive, as the complete circle is \(2\pi\) radians.

By understanding trigonometry in radians, you can solve complex problems, such as finding angles, calculating lengths, or modeling real-world phenomena with periodicity and oscillation.