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Chapter 6: Problem 54

Geometry If central angle \(\theta\) cuts off a chord of length \(c\) in a circle of radius \(\boldsymbol{r}\) (Figure 9), then the relationship between \(\theta, c\), and \(r\) is given by $$ 2 r \sin \frac{\theta}{2}=c $$ Find \(\theta\), if \(c=\sqrt{3} r\).

Short Answer

Expert verified
\( \theta = 120^\circ \) or \( \frac{2\pi}{3} \) radians.

Step by step solution

01

Understanding the Given Equation

We start with the equation that relates the central angle \( \theta \), the chord length \( c \), and the circle radius \( r \): \[ 2r \sin \frac{\theta}{2} = c \]. Our task is to find the value of \( \theta \) when \( c = \sqrt{3} r \).
02

Substituting the Given Values

Substitute \( c = \sqrt{3} r \) into the equation: \[ 2r \sin \frac{\theta}{2} = \sqrt{3} r \]. This simplifies to \[ 2 \sin \frac{\theta}{2} = \sqrt{3} \], since the \( r \) terms cancel each other out.
03

Solving for Sine

We now solve for \( \sin \frac{\theta}{2} \) by dividing both sides by 2: \[ \sin \frac{\theta}{2} = \frac{\sqrt{3}}{2} \].
04

Finding the Angle

The value \( \sin x = \frac{\sqrt{3}}{2} \) corresponds to angles \( x \) of \( 60^\circ \) or \( \frac{\pi}{3} \) radians (in one full rotation, assuming the angle is in the first quadrant). Hence, \( \frac{\theta}{2} = 60^\circ \).
05

Calculating \( \theta \)

Since \( \frac{\theta}{2} = 60^\circ \), we multiply by 2 to solve for \( \theta \): \( \theta = 2 \times 60^\circ = 120^\circ \). In radians, \( \theta = 2 \times \frac{\pi}{3} = \frac{2\pi}{3} \) radians.

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Key Concepts

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

Central Angle
In the world of circles, the _central angle_ is an angle whose vertex is the center of the circle and whose sides extend to the circumference. It effectively divides the circle into two arcs.
  • This angle is significant because it helps determine other properties within the circle, such as arc length and chord length.
  • In trigonometry, the central angle is often used in equations involving chords and arcs.
In our exercise, the central angle \( \theta \) is part of the equation \( 2r \sin \frac{\theta}{2} = c \), where \( c \) is the chord length. By solving this equation, you can determine the angle if certain values are known. This relationship highlights the interconnected nature of the parts of a circle.
Chord Length
The _chord length_ is the straight line distance between two points on a circle's circumference. It plays a vital role in circle geometry.
  • The chord divides the circle into segments and helps in calculating areas or central angles associated with those segments.
  • In our context, replacing the chord length \( c \) with \( \sqrt{3} r \) simplifies the equation to \( 2 \sin \frac{\theta}{2} = \sqrt{3} \).
  • This suggests that chord length is directly dependent on both the central angle and the circle鈥檚 radius.
Analyzing the chord length within a circle can reveal more about the geometric properties of the circle itself, especially when combined with the sine function to create equations.
Circle Radius
The _radius_ is the distance from the center of a circle to any point on its circumference. It is a fundamental characteristic vital to defining the size of a circle.
  • This simple measure is central to various circle-related calculations, including area and perimeter (circumference).
  • When paired with the central angle, as in \( 2r \sin \frac{\theta}{2} = c \), it assists in defining the relationship with chord length.
  • In our equation, the radius \( r \) cancels out, emphasizing the preeminence of angular properties and ratios.
Exploring how the circle's radius interacts with other circle properties is essential in understanding both basic and advanced geometrical concepts.
Sine Function
The _sine function_ is a fundamental concept in trigonometry and explores the relationship between different parts of a triangle.
  • In our equation, \( \sin \frac{\theta}{2} = \frac{\sqrt{3}}{2} \), this trigonometric function connects the central angle and the chord length.
  • The sine function provides a measure of the vertical component of the 'opposite' in a right triangle compared to the hypotenuse.
  • Solving for the angle using the sine function involves recognizing specific trigonometric values, such as \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \).
Thus, the sine function is a powerful tool in determining angles and their associated values, crucial in many areas of mathematics and engineering.

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Most popular questions from this chapter

For each of the following equations, solve for (a) all radian solutions and (b) \(t\) if \(0 \leq t<2 \pi\). Give all answers as exact values in radians. Do not use a calculator. $$ 3 \sin t+4=4 $$

For each of the following equations, solve for (a) all radian solutions and (b) \(t\) if \(0 \leq t<2 \pi\). Give all answers as exact values in radians. Do not use a calculator. $$ 5 \cos t+\sqrt{12}=\cos t $$

For each of the following equations, solve for (a) all radian solutions and (b) \(x\) if \(0 \leq x<2 \pi\). Give all answers as exact values in radians. Do not use a calculator. $$ \tan x(\tan x+1)=0 $$

\(2 \sin ^{2} 4 \theta-2 \cos 4 \theta=1\)

\(\cos 3 \theta=0\)
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