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Chapter 4: Problem 81

Find all real numbers in the interval \([0,2 \pi]\) that satisfy each equation. $$ \cos (x) \cos (-2 x)-\sin (x) \sin (-2 x)=\frac{\sqrt{3}}{2} $$

Short Answer

Expert verified
x = \[ \text{pi}/6, 11\text{pi}/6 \].

Step by step solution

01

- Use Angle Sum Identity

Recognize that the left-hand side of the equation can be written using the cosine angle sum identity: \[ \text{cos}(A+B)=\text{cos}(A)\text{cos}(B)-\text{sin}(A)\text{sin}(B) \] Here, let \( A = x \) and \( B = -2x \). Thus, the equation becomes: \[ \text{cos}(x - 2x) = \text{cos}(-x) = \frac{\text{sqrt{3}}}{2} \]
02

- Simplify the cosine of negative angle

Use the even property of cosine, which states that \( \text{cos}(-x) = \text{cos}(x) \), the equation simplifies to: \[ \text{cos}(x) = \frac{\text{sqrt{3}}}{2} \]
03

- Solve for x

Determine the values of \(x\) in the interval \[0, 2\text{pi}) \] that satisfy the equation \[ \text{cos}(x) = \frac{\text{sqrt{3}}}{2}. \] The specific angles are \[ \text{pi}/6 \] and \[ \text{11\text{pi}/6} \].

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Key Concepts

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

cosine angle sum identity
The cosine angle sum identity is a useful formula in trigonometry. It allows us to express the cosine of a sum of two angles, \((A + B)\), in terms of the cosines and sines of those angles individually. The identity is given by: \[ \text{cos}(A+B)=\text{cos}(A)\text{cos}(B)-\text{sin}(A)\text{sin}(B) \] In the original exercise, we apply this identity to the equation \(\text{cos}(x) \text{cos}(-2x) - \text{sin}(x) \text{sin}(-2x) = \frac{\text{sqrt{3}}{2}\). By letting \A = x\ and \B = -2x\, we transform the left-hand side to \(\text{cos}(x - 2x)\). This simplifies to \(\text{cos}(-x)\). This simplification is crucial for making the equation easier to solve.
cosine of negative angle
The property of the cosine of a negative angle is one of the fundamental aspects of trigonometric functions. The cosine function is 'even', which means: \[ \text{cos}(-x) = \text{cos}(x) \] This property aids in simplifying trigonometric equations. In our exercise, after recognizing that \(\text{cos}(x - 2x) = \text{cos}(-x)\), we apply the property of cosine being even. This simplifies \(\text{cos}(-x)\) to \(\text{cos}(x)\), making the equation more straightforward: \[ \text{cos}(x) = \frac{\text{sqrt{3}}{2}} \]
solving trigonometric equations
To solve trigonometric equations, follow these steps:
  • Firstly, simplify the equation using known identities, like the angle sum identity or the properties of trigonometric functions.
  • Next, rewrite the equation in a standard trigonometric form, such as \( \text{cos}(x) = y \) or \( \text{sin}(x) = y \).
  • Finally, identify the angle(s) that satisfy the equation within the given interval.
In this exercise, once we simplified the equation to \( \text{cos}(x) = \frac{\text{sqrt{3}}{2}}\), we know that \( x \) must be the angles where cosine equals \( \frac{\text{sqrt{3}}}{2} \). These angles within the interval \( [0, 2\text{pi}] \) are:
  • \( \text{pi}/6 \)
  • \( \text{11pi}/6 \)
Checking these angles confirms they satisfy the original equation.

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Most popular questions from this chapter

Solve each equation. Round approximate answers to the nearest tenth of a degree. $$ \begin{array}{l} (3.6)^{2}=(5.4)^{2}+(8.2)^{2}-2(5.4)(8.2) \cos \alpha \\ \text { for } 0^{\circ}<\alpha<90^{\circ} \end{array} $$

Find the exact value of each composition without using a calculator or table. $$ \arccos (\cos (-\pi / 3)) $$

Find the exact value of each composition without using a calculator or table. $$ \sin ^{-1}(\cos (2 \pi / 3)) $$

Find the exact value of each composition without using a calculator or table. $$ \cot (\operatorname{arccot}(0)) $$

Find all real numbers that satisfy each equation. Round approximate answers to 2 decimal places. $$ (4.1)^{2}=(2.4)^{2}+(5.3)^{2}-2(2.4)(5.3) \cos \alpha $$
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