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Chapter 6: Problem 27

\(\cos 2 x \cos x-\sin 2 x \sin x=-\frac{\sqrt{3}}{2}\)

Short Answer

Expert verified
Use double angle formulas and solve for x.

Step by step solution

01

Use Double Angle Formulas

Express \(\cos 2x\) and \(\sin 2x\) using their double angle identities: \[\cos 2x = \cos^2 x - \sin^2 x\] \[\sin 2x = 2 \sin x \cos x\].This will transform the initial equation into:\[(\cos^2 x - \sin^2 x) \cos x - (2 \sin x \cos x) \sin x = -\frac{\sqrt{3}}{2}\].

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Key Concepts

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

Cosine Double Angle Identity
The cosine double angle identity is an essential tool in trigonometry. It describes
  • how the cosine of twice an angle can be calculated using simpler expressions.
  • multiple forms, with the most common being:
\[\cos 2x = \cos^2 x - \sin^2 x\]Another useful variant is:\[\cos 2x = 2\cos^2 x - 1\]And yet another form is:\[\cos 2x = 1 - 2\sin^2 x\]These forms allow flexibility in solving different types of trigonometric equations.
To use the double angle formula effectively:
  • Identify the structure of your equation.
  • Select the form that simplifies your equation best.
When we look at these identities, they provide a powerful tool to express complicated trigonometric expressions in simpler terms, aiding in solving equations more efficiently. Remember to choose the identity that best fits the equation you're working with to simplify your solving process.
Sine Double Angle Identity
The sine double angle identity offers a way to express
  • the sine of twice an angle in terms of the sine and cosine of the original angle.
  • a simple relationship that is often used in trigonometric transformations:
\[\sin 2x = 2 \sin x \cos x\]This identity shows that the double angle sine is calculated by multiplying
  • twice the original angle's sine by its cosine.
  • a single term that involves both sine and cosine.
This formula is particularly useful for converting expressions that include sine and cosine functions of the same angle.
In exercises where you need to solve for variables or simplify equations, converting complex trigonometric expressions using the
  • sine double angle identity can provide a clear path to the solution.
Make sure to recognize when expressions can be converted with this identity to streamline your solving process.
Trigonometric Equations
Trigonometric equations are mathematical statements
  • that set trigonometric expressions equal to each other or to constant values.
  • present challenges that often involve finding unknown angles or proving identities.
To effectively solve trigonometric equations:
  • Begin by simplifying the equation using identities like the double angle formulas.
  • Identify any familiar or standard forms of the equation.
  • Use algebraic manipulation to isolate the trigonometric function.
  • Consider the domain and range of the trigonometric functions involved.
This specific approach helps determine the correct solutions, as most trigonometric functions are periodic, meaning there are often multiple valid solutions within a given range.
The ability to convert complex expressions into simpler equivalents via identities is crucial, especially when expressions can initially seem daunting or convoluted.
Through practice and understanding, trigonometric equations become manageable puzzles that link trigonometric concepts with algebraic techniques.

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Most popular questions from this chapter

\(\sin ^{3} 5 x=-1\)

\(\cot ^{2} 3 \theta=1\)

Use the quadratic formula to find (a) all degree solutions and (b) \(\theta\) if \(0^{\circ} \leq \theta<360^{\circ}\). Use a calculator to approximate all answers to the nearest tenth of a degree. $$ 1-4 \cos \theta=-2 \cos ^{2} \theta $$

\(\cos 8 \theta=\frac{1}{2}\)

For each of the following equations, solve for (a) all radian solutions and (b) \(x\) if \(0 \leq x<2 \pi\). Give all answers as exact values in radians. Do not use a calculator. $$ \tan x(\tan x-1)=0 $$
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