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Chapter 8: Problem 6

Express the given angle measurements in radian measure in terms of \(\pi\). $$36^{\circ}, 315^{\circ}$$

Short Answer

Expert verified
\(36^{\circ}\) is \(\frac{\pi}{5}\) radians, and \(315^{\circ}\) is \(\frac{7\pi}{4}\) radians.

Step by step solution

01

Understand the Conversion Formula

To convert degrees to radians, we use the conversion formula \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). This relates degrees and radians using the fact that a full circle is \(360^{\circ}\) or \(2\pi\) radians, which simplifies to \(1^{\circ} = \frac{\pi}{180}\) radians.
02

Convert 36 Degrees to Radians

Apply the conversion formula for \(36^{\circ}\):\[36^{\circ} \times \frac{\pi}{180} = \frac{36\pi}{180} = \frac{\pi}{5}\]Hence, \(36^{\circ}\) is equivalent to \(\frac{\pi}{5}\) radians.
03

Convert 315 Degrees to Radians

Now, apply the conversion formula for \(315^{\circ}\):\[315^{\circ} \times \frac{\pi}{180} = \frac{315\pi}{180} = \frac{7\pi}{4}\]Thus, \(315^{\circ}\) is equivalent to \(\frac{7\pi}{4}\) radians.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radian Measure
The radian measure is a way of expressing angles that is widely used in mathematics, particularly in trigonometry and calculus. Radians offer a more natural approach than degrees for many calculations because they are based on the radius of a circle. The concept is simple: an angle's radian measure is the length of the arc of a circle of radius 1 (a unit circle) subtended by the angle. A full circle, therefore, measures about 6.283 radians, which we often approximate as \[ 2\pi \text{ radians} \]. Similarly, half a circle (a straight line) is \( \pi \) radians.
  • This natural unit is perfect for continuous functions, providing a direct link between angular and linear measurements.
  • Unlike degrees, which divide a circle into 360 parts, radians provide a seamless connection with the geometry of circles.
Understanding radian measure can greatly enhance your ability to work with angles in many branches of mathematics.
Degree to Radian Conversion
Converting degrees to radians is a fundamental yet straightforward skill in trigonometry and geometry.The conversion factor between degrees and radians is crucial: \[1^{\circ} = \frac{\pi}{180} \text{ radians}.\]This relationship arises because a full circle of 360 degrees is equal to \( 2\pi \) radians.Here's a quick way to remember it:
  • Multiply degrees by \( \frac{\pi}{180} \) to convert to radians.
  • The fraction \( \frac{\pi}{180} \) stems from dividing the \(2\pi\) radians of a full circle by 360 degrees.
This process allows for the flexible switching between degree and radian measures, aiding in diverse mathematical tasks.By mastering the degree to radian conversion, you enhance your versatility in mathematical, scientific, and engineering calculations.
Pi
Pi (\(\pi\)) is an essential mathematical constant representing the ratio of any circle's circumference to its diameter.
  • Pi is an irrational number, approximately equal to 3.14159, and it continues infinitely without repeating.
  • It appears in various mathematical formulas and is particularly vital in trigonometry for angle measurements in radians.
The appearance of \(\pi\) in the context of radian measure and angle conversion is natural and intuitive since it arises from the circle's properties.When we convert degrees to radians, \(\pi\) helps to express the ratio of a given angle to the whole circle. For example, \(180^{\circ}\) equals \(\pi\) radians because a semi-circle is a half of \(2\pi\), the full circle's radian measure.Understanding \(\pi\) is crucial for diving deeper into topics like trigonometry and calculus, where circles and periodic functions play a key role.
Trigonometry
Trigonometry is a branch of mathematics dealing with the relationships between the angles and sides of triangles.
  • It extends these concepts to the unit circle, facilitating calculations involving circular and oscillatory motion.
  • The basic trigonometric functions鈥攕ine, cosine, and tangent鈥攄epend inherently on angles measured in radians.
Utilizing radians in trigonometry allows for simpler derivatives and integrals in calculus. For instance, when you use radians, the derivative of \( \sin(x) \) is \( \cos(x) \), which shows how naturally these functions behave in their radian form.Thus, understanding the radian measure enhances your trigonometric skills, paving the way for more advanced mathematical concepts.Trigonometry's applications stretch far beyond solving triangles, making it fundamental for fields such as physics, engineering, and any discipline involving waves or periodic phenomena.

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