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Chapter 5: Problem 72

Find exact solutions, where \(0 \leq x<2 \pi\) $$\cos 2 x \cos x+\sin 2 x \sin x=-1$$

Short Answer

Expert verified
The exact solution of the given equation within the given interval is \(x = \pi\).

Step by step solution

01

Use of Trigonometric Identity

We have \(\cos(2x) \cos(x) + \sin(2x) \sin(x)\), by using the trigonometric identity \(\cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)\), we can write the equation as \(\cos(2x - x) = -1\). The given equation simplifies to \(\cos(x) = -1\).
02

Solve for x

We need to find the solutions of the equation \(\cos(x) = -1\), within the given range \(0 \leq x < 2\pi\). The function \(\cos(x)\) equals -1 at x = \(\pi\) within the given interval.
03

Check the solution

We have found the solution x = \(\pi\). Plug this value back into the original equation to verify that the left-hand side equals the right-hand side.

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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 is a fundamental concept in trigonometry, often represented as \( \cos(\theta) \), where \( \theta \) is an angle measured in radians or degrees. It describes the relationship between the angle and the length of the adjacent side over the hypotenuse in a right-angled triangle.

Some key points to remember about the cosine function are:
  • It is a periodic function with a period of \( 2\pi \).
  • The range of the cosine function is from -1 to 1.
  • \( \cos(0) = 1 \) and \( \cos(\pi) = -1 \).


Understanding the behavior of the cosine function, such as when it reaches its maximum, minimum, or zero values, is crucial when solving trigonometric equations. For instance, in our problem, determining when \( \cos(x) = -1 \) helped find the solution \( x = \pi \).
Sine Function
The sine function is another primary function in trigonometry, noted as \( \sin(\theta) \). It represents the ratio of the length of the opposite side to the hypotenuse in a right triangle. Similar to the cosine function, the sine function demonstrates periodic properties.

Here are some important characteristics of the sine function:
  • It has a period of \( 2\pi \).
  • The sine function ranges from -1 to 1.
  • \( \sin(0) = 0 \), and \( \sin(\pi/2) = 1 \).


In many problems, including the one provided, understanding how to manipulate sine in conjunction with cosine is key. In our exercise, the sine and cosine functions were used together to express the equation in a more manageable form by employing trigonometric identities.
Angle Subtraction Identity
The angle subtraction identity is a powerful tool in trigonometry. It allows simplification of complex expressions involving trigonometric functions. The identity states:

\[ \cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta) \]

This identity is particularly useful for combining or reducing angles in expressions.
  • It simplifies solving trigonometric equations.
  • Helps convert sums and differences of angles into single angles.


In the original exercise, this identity was used to simplify \( \cos(2x)\cos(x) + \sin(2x)\sin(x) \) into \( \cos((2x - x)) = \cos(x) \), making it easier to solve for \( x \).

Recognizing when to apply identities like angle subtraction is a critical skill in trigonometry, aiding in both simplification and solution of complex problems.

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