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Chapter 8: Problem 1

(I) Express the following angles in radians: \((a)\) 45.0\(^{\circ}\), \((b)\) 60.0\(^{\circ}\), \((c)\) 90.0\(^{\circ}\), \((d)\) 360.0\(^{\circ}\), and \((e)\) 445\(^{\circ}\). Give as numerical values and as fractions of \(\pi\).

Short Answer

Expert verified
45掳 = 蟺/4, 60掳 = 蟺/3, 90掳 = 蟺/2, 360掳 = 2蟺, 445掳 = 89蟺/36 radians.

Step by step solution

01

Understanding Degrees to Radians Conversion

To convert degrees to radians, we use the formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). This formula allows us to express any angle measured in degrees as an angle in radians.
02

Convert 45 Degrees to Radians

Using the formula, we substitute 45 degrees: \( 45 \times \frac{\pi}{180} = \frac{\pi}{4} \). Numerically, \( \frac{\pi}{4} \approx 0.785 \) radians.
03

Convert 60 Degrees to Radians

Substitute 60 degrees into the formula: \( 60 \times \frac{\pi}{180} = \frac{\pi}{3} \). This is approximately \( 1.047 \) radians.
04

Convert 90 Degrees to Radians

Using the conversion formula: \( 90 \times \frac{\pi}{180} = \frac{\pi}{2} \), which numerically is \( 1.571 \) radians.
05

Convert 360 Degrees to Radians

Substitute 360 degrees: \( 360 \times \frac{\pi}{180} = 2\pi \). This conversion is equivalent to \( 6.283 \) radians.
06

Convert 445 Degrees to Radians

For 445 degrees, the calculation is: \( 445 \times \frac{\pi}{180} = \frac{445\pi}{180} \). Simplified, \( \frac{445\pi}{180} = \frac{89\pi}{36} \), approximately \( 7.76 \) radians.

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Key Concepts

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

Degrees to Radians Conversion
Converting degrees to radians is a fundamental skill in trigonometry and is essential for understanding angle measurements in different formats. The basic formula for this conversion is \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). This equation stems from the fact that a full circle is 360 degrees or \( 2\pi \) radians.Here's a quick breakdown:
  • One complete circle or rotation is 360 degrees and is equivalent to \( 2\pi \) radians.
  • To find radians from degrees, multiply the degree value by \( \frac{\pi}{180} \).
For example, converting 45 degrees to radians involves multiplying by this fraction, resulting in \( \frac{\pi}{4} \). This conversion helps bridge the gap between different systems of angle measurement and is particularly useful in mathematical equations and studies involving circles.
Angle Measurement
Angle measurement can be expressed in degrees or radians, and understanding this can help in various mathematical and practical applications. Degrees are typically used in everyday contexts, such as in navigation or in geometry classrooms, while radians are more common in advanced mathematics and physics.Key points about angle measurement methods:
  • A degree is 1/360th of a circle, making it a familiar unit for common geometrical work.
  • A radian, however, is based on the radius of the circle, where the angle subtended by an arc equal in length to the radius is one radian.
  • In terms of comparison, \( 180^{\circ} = \pi \) radians, thus 1 radian \( \approx 57.2958^{\circ} \).
Mastering these concepts allows one to seamlessly transition between using degrees and radians depending on the requirements of a mathematical problem or real-life application.
Trigonometry
Trigonometry primarily deals with the study of triangles, and especially right-angled triangles. It revolves around the relationships between angle measures and lengths of triangle sides, using functions such as sine, cosine, and tangent. Why radians are often preferred in trigonometry:
  • Radians simplify derivative and integral calculations in calculus, as trigonometric functions' derivatives and integrals are much more straightforward when angles are in radians.
  • Many trigonometric identities and formulas are naturally derived and simpler in radian measurement.
  • Using radians can make trigonometric graphs more intuitive, as the period of function graphs align more naturally with radian measure.
Ultimately, understanding radians is crucial for deeper exploration of trigonometric concepts, as well as their applications in wave functions, signals, and oscillatory motion found in physics.

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Most popular questions from this chapter

(I) The Sun subtends an angle of about 0.5\(^{\circ}\) to us on Earth, 150 million km away. Estimate the radius of the Sun.

(II) How fast (in rpm) must a centrifuge rotate if a particle 8.0 cm from the axis of rotation is to experience an acceleration of 100,000 g's?

(III) A 1.80-m-long pole is balanced vertically with its tip on the ground. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [\(Hint\): Use conservation of energy.]

(II) A person exerts a horizontal force of 42 N on the end of a door 96 cm wide. What is the magnitude of the torque if the force is exerted \((a)\) perpendicular to the door and \((b)\) at a 60.0\(^{\circ}\) angle to the face of the door?

(I) Pilots can be tested for the stresses of flying high-speed jets in a whirling "human centrifuge," which takes 1.0 min to turn through 23 complete revolutions before reaching its final speed. \((a)\) What was its angular acceleration (assumed constant), and \((b)\) what was its final angular speed in rpm?
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