Problem 51 Solve the given equation. $$\c... [FREE SOLUTION]

Chapter 9: Problem 51

Solve the given equation. $$\cos \theta(2 \sin \theta+1)=0$$

Short Answer

Expert verified

$ \theta = \frac{\pi}{2} + n\pi $ and $ \theta = \frac{7\pi}{6} + 2m\pi $, $ \theta = \frac{11\pi}{6} + 2m\pi $, where $ n, m $ are integers.

Step by step solution

Set Each Factor to Zero

The equation $ \cos \theta (2 \sin \theta + 1) = 0 $ can be solved by setting each factor equal to zero. This gives us two separate equations: $ \cos \theta = 0 $ and $ 2 \sin \theta + 1 = 0 $.

Solve for $ \theta $ from $ \cos \theta = 0 $

The cosine function equals zero at $ \theta = \frac{\pi}{2} $ and $ \theta = \frac{3\pi}{2} $ within one full circle (0 to $2\pi$). Thus, $ \theta = \frac{\pi}{2} + n\pi $, where $ n $ is an integer.

Solve for $ \theta $ from $ 2 \sin \theta + 1 = 0 $

Rearrange the equation $ 2 \sin \theta + 1 = 0 $ to solve for sine: $ 2 \sin \theta = -1 $ leading to $ \sin \theta = -\frac{1}{2} $. The sine function equals $-\frac{1}{2}$ at $ \theta = \frac{7\pi}{6} $ and $ \theta = \frac{11\pi}{6} $ within one circle.

Combine the Solutions

The complete set of solutions for $ \theta $ includes all angles where either $ \cos \theta = 0 $ or $ \sin \theta = -\frac{1}{2} $. Therefore, $ \theta = \frac{\pi}{2} + n\pi $ and $ \theta = \frac{7\pi}{6} + 2m\pi $ or $ \theta = \frac{11\pi}{6} + 2m\pi $, where $ n $ and $ m $ are integers.

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Key Concepts

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

Cosine Function

The cosine function, often written as $ \cos \theta $, is a fundamental trigonometric function that represents the x-coordinate of a point on the unit circle. For any angle $ \theta $, it measures how far along the horizontal axis the point corresponding to $ \theta $ is. Here's a breakdown of the cosine function:

Periodicity: The cosine function is periodic with a period of $2\pi$. This means its values repeat every $2\pi$ radians.
Range: The values of $ \cos \theta $ always lie between -1 and 1 inclusive, i.e., $-1 \leq \cos \theta \leq 1$.
Zeros: The cosine function equals zero at specific angles: $ \frac{\pi}{2}, \frac{3\pi}{2}, \ldots, \frac{\pi}{2} + n\pi $, where $ n $ is an integer.

In solving equations involving $ \cos \theta $, identifying points where the function is zero is crucial. This knowledge helps find specific solutions within a given interval, such as $ \theta = \frac{\pi}{2} $ or $ \frac{3\pi}{2} $ when solving for $ \cos \theta = 0 $. Understanding the pattern of zeros helps us solve trigonometric equations efficiently.

Sine Function

The sine function, or $ \sin \theta $, is another core trigonometric function that represents the y-coordinate of a point on the unit circle. It helps determine the vertical position of the point corresponding to an angle $ \theta $. Key features include:

Periodicity: Like the cosine, the sine function repeats every $2\pi$ radians.
Range: The values of $ \sin \theta $ are also between -1 and 1, i.e., $-1 \leq \sin \theta \leq 1$.
Zeros: The sine function becomes zero at $ \theta = n\pi $, where $ n $ is any integer. This occurs at angles like $ 0 $, $ \pi $, and $ 2\pi $.

When solving the equation $ 2 \sin \theta + 1 = 0 $, it's first transformed to $ \sin \theta = -\frac{1}{2} $. By understanding where sine takes this value, specifically at angles like $ \frac{7\pi}{6} $ and $ \frac{11\pi}{6} $, we can pinpoint the correct solutions. These angles, within one revolution of the circle, guide us to solve the trigonometric equation effectively.

Solving Equations

Trigonometric equations often require isolating and solving two factors separately, as illustrated by the equation $ \cos \theta(2 \sin \theta+1)=0 $. Here鈥檚 how to tackle such equations:

Factoring: The initial step is to identify and set each factor to zero. This transforms the problem into simpler components: $ \cos \theta = 0 $ and $ 2 \sin \theta + 1 = 0 $.
Solving separate equations: Each equation is solved for $ \theta $ separately, considering the periodic nature of trigonometric functions.
Combining solutions: After solving the individual equations within a specified interval (usually $ 0 $ to $ 2\pi $), combine all solutions to form a comprehensive set. This ensures all possible $ \theta $ values that satisfy the original equation are included.

The strategy of solving trigonometric equations by splitting them into simpler parts and using known values of sine or cosine at specific angles is a powerful tool. This approach allows for systematic problem-solving and ensures that no solution is missed.

91影视

Solve the given equation. $$\cos \theta(2 \sin \theta+1)=0$$

Short Answer

Step by step solution

Set Each Factor to Zero

Solve for \( \theta \) from \( \cos \theta = 0 \)

Solve for \( \theta \) from \( 2 \sin \theta + 1 = 0 \)

Combine the Solutions

Key Concepts

Cosine Function

Sine Function

Solving Equations

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