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Chapter 6: Problem 2

Find all solutions on the interval \(0 \leq \theta<2 \pi\). $$ 2 \sin (\theta)=\sqrt{3} $$

Short Answer

Expert verified
The solutions are \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{2\pi}{3} \).

Step by step solution

01

Isolate the Trigonometric Function

Start with the equation \( 2 \sin(\theta) = \sqrt{3} \). Divide both sides by 2 to solve for \( \sin(\theta) \):\[ \sin(\theta) = \frac{\sqrt{3}}{2} \].
02

Identify Angles with Known Sine Values

Determine which angle(s) \( \theta \) produces \( \sin(\theta) = \frac{\sqrt{3}}{2} \). Know that \( \sin(\theta) = \frac{\sqrt{3}}{2} \) at \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{2\pi}{3} \) in the interval \(0 \leq \theta < 2\pi\).
03

Confirm the Range

Since the problem specifies the interval \(0 \leq \theta < 2\pi\), check that both \( \frac{\pi}{3} \) and \( \frac{2\pi}{3} \) fall within this range, which they do.

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Key Concepts

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

Sine Function
The sine function is a fundamental concept in trigonometry, representing a periodic waveform. It is denoted by the symbol \( \sin \) and relates the angle of a right-angled triangle to the ratio of the length of the opposite side to the hypotenuse. This ratio results in a value between -1 and 1 for any given angle.

The sine function has several key properties:
  • It is periodic with a period of \( 2\pi \), meaning that \( \sin(\theta + 2\pi) = \sin(\theta) \).
  • The function is symmetric about the origin, which implies \( \sin(-\theta) = -\sin(\theta) \).
  • It reaches a maximum value of 1 and a minimum of -1.
Understanding these properties is crucial for solving trigonometric equations like the one in the exercise. Once the sine value for a specific angle is known, it can be used to identify other angles with the same sine value.
Angle Identification
In trigonometric equations, identifying angles that satisfy a certain condition, like a specific sine value, is essential. The unit circle represents angles and their sine values, providing a visual aid in identifying these angles.

For example, in the given equation \( 2 \sin(\theta) = \sqrt{3} \), we solve for \( \sin(\theta) = \frac{\sqrt{3}}{2} \). The unit circle reveals that this value corresponds to specific angles, namely \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{2\pi}{3} \).

Useful tips for angle identification include:
  • Familiarize yourself with common sine values and their corresponding angles.
  • Use symmetry properties of the sine function; angles with the same sine value can be found in symmetrical positions about the y-axis.
  • Recall that sine values repeat every \( 2\pi \), understanding this periodicity can help find all possible solutions.
By using these methods, one can efficiently determine all relevant angles that satisfy the given sine condition.
Trigonometric Interval Solutions
Trigonometric interval solutions refer to finding all possible angle measurements within a specified range that satisfy a given trigonometric equation. Such intervals are crucial for practical applications, ensuring results are limited to relevant domains.

For the equation \( 2 \sin(\theta) = \sqrt{3} \), we seek solutions in the interval \( 0 \leq \theta < 2\pi \). This means we need angles within one full rotation on the unit circle.

Here's how to approach interval solutions:
  • Check that identified angles, \( \frac{\pi}{3} \) and \( \frac{2\pi}{3} \), fall within the interval.
  • If angles exceed the specified interval, consider using equivalent angles via periodicity by adding or subtracting \( 2\pi \) as needed.
  • Remember to include only angles that strictly satisfy the interval condition.
This structured approach ensures that all solutions are both correct and applicable within the given constraints, making it easier to tackle complex trigonometric problems.

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Most popular questions from this chapter

Find all solutions. $$ 2 \sin (\theta)=-\sqrt{3} $$

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