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Chapter 8: Problem 13

Solve each equation on the interval \(0 \leq \theta<2 \pi\). \(2 \sin \theta+3=2\)

Short Answer

Expert verified
The solutions are \(\theta = \frac{7 \pi}{6}\) and \(\theta = \frac{11 \pi}{6}\).

Step by step solution

01

Isolate the sine function

Subtract 3 from both sides of the equation: \(2 \sin \theta + 3 - 3 = 2 - 3\) This simplifies to: \(2 \sin \theta = -1\)
02

Solve for \( \sin \theta\)

Divide both sides by 2 to isolate \( \sin \theta\): \(\sin \theta = \frac{-1}{2}\)
03

Determine the angles that satisfy the equation

Find values of \( \theta\) in the given interval \(0 \leq \theta < 2\pi\) where \(\sin \theta = -1/2\). These angles are: \(\theta = \frac{7 \pi}{6}\) and \(\theta = \frac{11 \pi}{6}\) because in those points the sine function equals \(-1/2\).

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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 one of the fundamental trigonometric functions. It relates to the y-coordinate of a point on the unit circle. For an angle \(\theta\), the sine function \(\sin \theta\) represents the height of that point from the origin. In trigonometry, understanding the sine function's behavior is key to solving various problems. The function repeats its values in a pattern every \(2\pi\), meaning \(\sin(\theta + 2\pi) = \sin\theta\). Knowing this periodic behavior helps in finding multiple solutions within a given interval.
interval notation
Interval notation is a way of representing a set of numbers between two endpoints. In this exercise, we are working with the interval \(0 \leq \theta \lt 2\pi\), which indicates all the values of \(\theta\) between 0 and \(2\pi\) (including 0 but not \(2\pi\)). This interval encompasses one complete cycle of the trigonometric functions, making it a standard range for problems involving trigonometric equations or identities. By using interval notation, we can concisely specify the domain in which we need to find the solutions.
trigonometric identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variable(s) where the functions are defined. They are essential tools for simplifying and solving trigonometric equations. Common identities include the Pythagorean identity (\(\sin^2 \theta + \cos^2 \theta = 1\)), angle sum and difference identities, and double-angle identities. When you encounter complex trigonometric equations, using these identities can often simplify the equation, making it easier to find solutions. In our exercise, we primarily use the property of the sine function to identify solutions.
solving equations
Solving equations involves finding all values of the variable that make the equation true. For trigonometric equations, like the one in our exercise, we use several steps:
  • Isolating the trigonometric function
  • Solving for the basic angle
  • Finding all angles within the given interval that satisfy the equation
Let's review our given problem. Initially, we isolate the sine function: \(2 \sin \theta + 3 - 3 = 2 - 3\). This simplifies to \(2 \sin \theta = -1\). To solve for \sin \theta, we divide both sides by 2, getting \(\sin \theta = \frac{-1}{2}\). Finally, we determine which angles between 0 and \2\pi have sine values of \-1/2. These angles are \(\frac{7 \pi}{6}\) and \(\frac{11 \pi}{6}\). Understanding and following these steps will help you solve similar trigonometric equations.

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