Problem 134 - Precalculus

Question

In calculus we work with real numbers; thus, the measure of an angle must be in radians. Determine the angle of the smallest possible positive measure (in radians) that is coterminal with the angle \(750^{\circ}\).

Step-by-Step Solution

Answer

The smallest positive coterminal angle with \(750^{\circ}\) is \(\frac{13\pi}{6}\) radians.

Step 1: Convert Degrees to Radians

To convert an angle from degrees to radians, we use the conversion factor \( \frac{\pi}{180^{\circ}} \). Therefore, to convert \(750^{\circ}\) to radians, we compute:\[750^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{750\pi}{180} \]Simplifying, we get:\[\frac{750\pi}{180} = \frac{25\pi}{6} \]

Step 2: Find Coterminal Angle in Radians

Coterminal angles differ by multiples of \(2\pi\) radians. We need the smallest positive angle coterminal with \(\frac{25\pi}{6}\). Subtract \(2\pi\) repeatedly:\[\frac{25\pi}{6} - 2\pi = \frac{25\pi}{6} - \frac{12\pi}{6} = \frac{13\pi}{6} \]Since \(\frac{13\pi}{6} > 0\) and less than \(2\pi\), it's the smallest positive coterminal angle.

Key Concepts

Radian Measure

Understanding radian measure is key when working with angles, especially in calculus. Radians offer a more natural way to express angles in relation to circles. Unlike degrees, which divide a circle into 360 parts, radians relate directly to the circle's radius.

• One full revolution around a circle is equal to \(2\pi\) radians.
• This means that a half-circle, or \(180^{\circ}\), is \(\pi\) radians.

When converting angles from degrees to radians, use the conversion factor \(\frac{\pi}{180^{\circ}}\). This approach scales the degree measure into a radian measure by linking the segment of the circle's circumference to the radius. Radians create a tight bond between linear and angular measurements, which is useful for calculus and physics.

Angle Conversion

Converting angles from degrees to radians or vice versa is a foundational skill in trigonometry. For this, you will rely on the concepts of ratio and proportion.To convert from degrees to radians:1. Multiply the degree measure by \(\frac{\pi}{180^{\circ}}\). - For example, converting \(750^{\circ}\) results in: \[ 750^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{750\pi}{180} = \frac{25\pi}{6} \]2. Simplifying fractions in radian form helps in finding accurate measures for calculations.For converting radians to degrees:- Multiply the radian measure by \(\frac{180^{\circ}}{\pi}\).Conversion supports the use of radians in calculus, where trigonometric functions commonly use radian measure over degrees for their natural periodic properties.

Trigonometry

Trigonometry often deals with angles and their measurements, making the understanding of radian measure essential. It's pivotal when dealing with functions like sine, cosine, and tangent, which assume the angle input is in radians by default.

• With radians, angles have a linear relationship with arc length, simplifying calculations.
• Coterminal angles share the same terminal side when drawn in standard position on the coordinate plane. - They can be found by adding or subtracting \(2\pi\) radians to the given angle.

In the given problem, converting \(750^{\circ}\) to \(\frac{25\pi}{6}\) radians and finding a coterminal angle \(\frac{13\pi}{6}\) shows how radians seamlessly integrate into trigonometric applications. This understanding aids in solving complex problems and performing calculus efficiently.