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Chapter 12: Problem 65

\(4 \sin x \cos x-2 \cos x-2 \sqrt{3} \sin x+\sqrt{3}=0\)

Short Answer

Expert verified
The detailed calculation depends on the familiarity with trigonometric tables or the unit circle. The values of \(x\) would satisfy the original equation.

Step by step solution

01

Recognize and Apply Trigonometric Identities

Start by observing that the expression \(4 \sin x \cos x\) can be re-written in terms of double-angles. We use the double angle formula for sine \(2\sin x \cos x = \sin 2x\), thus simplifying the equation to: \(2\sin 2x - 2 \cos x - 2 \sqrt{3} \sin x + \sqrt{3} = 0\)
02

Isolate the terms with sine and cosine

Rearrange the terms to better isolate sine and cosine, the resulting equation becomes: \(2\sin 2x - 2 \sqrt{3} \sin x - 2 \cos x + \sqrt{3} = 0\)
03

Further simplifications

Divide the entire equation by 2 to simplify it: \(\sin 2x -\sqrt{3} \sin x - \cos x + \frac{\sqrt{3}}{2} = 0\)
04

Solve the equation

At this point, this trigonometric equation can be solved using common methods for solving trigonometric equations, which includes the use of unit circles or trigonometric tables to find the solutions

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Key Concepts

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

Trigonometric Identities
Trigonometric identities are formulas that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equality are defined. They are useful tools in simplifying equations and solving trigonometric problems. One of the commonly used identities is the double angle formula for sine:
  • \(2 \sin x \cos x = \sin 2x\)
  • This identity helps simplify expressions by converting products of sine and cosine into single terms involving double angles.
For example, in the given problem, the expression \(4 \sin x \cos x\) was simplified using this identity, making it easier to work with. Trigonometric identities also include basic ones like \(\sin^2 x + \cos^2 x = 1\), which can be handy in transforming equations.
Double Angle Formulas
Double angle formulas are specific types of trigonometric identities that express trigonometric functions of double angles (like \(2x\)) in terms of single angles \(x\). Here are the basic ones:
  • For sine: \(\sin 2x = 2\sin x \cos x\)
  • For cosine: \(\cos 2x = \cos^2 x - \sin^2 x\) or \(1 - 2\sin^2 x\) or \(2\cos^2 x - 1\)
  • For tangent: \(\tan 2x = \frac{2\tan x}{1 - \tan^2 x}\)
These formulas are incredibly helpful in simplifying trigonometric equations or expressions involving \(2x\). They allow you to express the function in terms of \(x\), which is easier to manage and solve, especially in complex equations. In our exercise, the formula for sine was used to transform a product \(4\sin x\cos x\) into \(2\sin 2x\), streamlining the equation.
Solving Trigonometric Equations
Solving trigonometric equations involves finding angle(s) \(x\) that satisfy the equation for known trigonometric values. Methods include using algebraic identities, graphical methods, or numerical solutions through calculators or software.
  • One common approach is to isolate trigonometric terms, often simplifying the equation into functions involving only sine, cosine, or tangent.
  • Next, use identities like the unit circle or reference angles to find values of \(x\) that solve the equation.
  • Because trigonometric functions are periodic, solutions often come in infinite sets, expressed with general solutions that include integer multiples of the function's period.
In the problem presented, after using identities and simplifying, we would rely on standard trigonometric solutions methods, such as checking corresponding angles in the unit circle, to find all potential solutions for \(x\).

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