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Chapter 6: Problem 32

The measure of an angle in standard position is given. Find two positive angles and two negative angles that are co terminal with the given angle. $$\frac{11 \pi}{6}$$

Short Answer

Expert verified
Positive: \( \frac{23\pi}{6}, \frac{35\pi}{6} \); Negative: \( -\frac{\pi}{6}, -\frac{13\pi}{6} \).

Step by step solution

01

Identify the Coterminal Angle Formula

Coterminal angles differ by complete rotations of the circle. The coterminal angles of a given angle \( \theta \) can be found by adding or subtracting \( 2\pi \), since one full rotation is equal to \( 2\pi \) in radians.
02

Find Two Positive Angles

To find the first positive coterminal angle of \( \frac{11\pi}{6} \), add \( 2\pi \). This gives:\[\frac{11\pi}{6} + \frac{12\pi}{6} = \frac{23\pi}{6}.\]Perform another addition:\[\frac{23\pi}{6} + \frac{12\pi}{6} = \frac{35\pi}{6}.\]Thus, the two positive coterminal angles are \( \frac{23\pi}{6} \) and \( \frac{35\pi}{6} \).
03

Find Two Negative Angles

To find the first negative coterminal angle of \( \frac{11\pi}{6} \), subtract \( 2\pi \). This gives:\[\frac{11\pi}{6} - \frac{12\pi}{6} = -\frac{\pi}{6}.\]Perform another subtraction:\[-\frac{\pi}{6} - \frac{12\pi}{6} = -\frac{13\pi}{6}.\]Thus, the two negative coterminal angles are \( -\frac{\pi}{6} \) and \( -\frac{13\pi}{6} \).

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Key Concepts

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

Radians
Radians are a way to measure angles that relate directly to the geometry of a circle. Instead of degrees, radians utilize the radius of the circle to define an angle's size. This forms a relationship between the length of the arc of the circle and the radius, making calculations often more straightforward and elegant in mathematics.

The entire circumference of a circle is connected with the radian measurement through the formula:
  • Circumference of a circle: \( 2\pi \cdot r \), where \( r \) is the radius.
  • Radians of a full circle: \( 2\pi \), where 1 full rotation equals \( 360^{\circ} \).
Hence, an angle measured in radians directly depicts how many radii will fit around the circle's arc encompassed by the angle.
Angle Measurement
Measuring angles is crucial, especially when working with rotations and circular movements. Angles can be measured in two primary units: degrees and radians. The choice of unit depends on the context, but radians are often more convenient for mathematical calculations.

In the radian system:
  • \( \pi \) radians = \( 180^{\circ} \)
  • \( 2\pi \) radians is equivalent to a full circle, which is \( 360^{\circ} \).
Understanding how these measurements interact is important for solving problems. For instance, being able to convert between degrees and radians allows you to utilize the most appropriate unit based on the problem's requirements.
Angle Addition and Subtraction
Angle addition and subtraction are key operations when dealing with angles, particularly with the concept of coterminal angles. Coterminal angles are angles that share the same terminal side but may have different measures. Understanding how to find them involves simple arithmetic operations.

To compute coterminal angles:
  • For positive coterminal angles, add \( 2\pi \) (a full rotation) to the given angle. Continue this process for multiple positive coterminals.
  • For negative coterminal angles, subtract \( 2\pi \) from the given angle. Repeat the subtraction for more negative coterminal angles.
These operations allow you to see not just one, but a continuous spectrum of angles that look identical in terms of their positions, differing only by the number of complete cycles they include.

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Most popular questions from this chapter

A truck with 48 -in.-diameter wheels is traveling at \(50 \mathrm{mi} / \mathrm{h}\). (a) Find the angular speed of the wheels in rad/min. (b) How many revolutions per minute do the wheels make?

Prove that in triangle \(A B C\) $$\begin{array}{l}a=b \cos C+c \cos B \\\b=c \cos A+a \cos C \\\c=a \cos B+b \cos A\end{array}$$ These are called the Projection Laws. IHint: To get the first equation, add the second and third equations in the Law of cosines and solve for \(a .]\)

Tracking a Satellite The path of a satellite orbiting the earth causes the satellite to pass directly over two tracking stations \(A\) and \(B,\) which are 50 mi apart. When the satellite is on one side of the two stations, the angles of elevation at \(A\) and \(B\) are measured to be \(87.0^{\circ}\) and \(84.2^{\circ},\) respectively. (a) How far is the satellite from station \(A ?\) (b) How high is the satellite above the ground? (Image can't copy)

Soap Bubbles When two bubbles cling together in midair, their common surface is part of a sphere whose center \(D\) lies on the line passing through the centers of the bubbles (see the figure). Also, \(\angle A C B\) and \(\angle A C D\) each have measure \(60^{\circ} .\) (a) Show that the radius \(r\) of the common face is given by $$ r=\frac{a b}{a-b} $$ [Hint: Use the Law of Sines together with the fact that an angle \(\theta\) and its supplement \(180^{\circ}-\theta\) have the same sine.] (b) Find the radius of the common face if the radii of the bubbles are \(4 \mathrm{cm}\) and \(3 \mathrm{cm} .\) (c) What shape does the common face take if the two bubbles have equal radii? (Image can't copy)

These exercises involve the formula for the area of a circular sector. The area of a sector of a circle with a central angle of \(5 \pi / 12\) rad is \(20 \mathrm{m}^{2} .\) Find the radius of the circle.
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