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Magazine Chapter 8: Problem 42
Establish each identity. $$ 1-\cos (2 \theta)+\cos (4 \theta)-\cos (6 \theta)=4 \sin \theta \cos (2 \theta) \sin (3 \theta) $$
Short Answer
Expert verified
LHS = RHS after simplifying trigonometric identities.
Step by step solution
01
- Understand the Problem
The goal is to establish that the left-hand side (LHS) of the equation is equal to the right-hand side (RHS). The LHS is given by: \(1 - \cos(2\theta) + \cos(4\theta) - \cos(6\theta)\) and the RHS is given by: \(4 \sin(\theta) \cos(2\theta) \sin(3\theta)\).
02
- Apply Trigonometric Identities
We need to use trigonometric identities to simplify the LHS. Start with the double angle formulas: \(\cos(2\theta) = 1 - 2\sin^2(\theta)\), \(\cos(4\theta)=2\cos^2(2\theta)-1\), and \(\cos(6\theta)=1-2\sin^2(3\theta)\).
03
- Rewrite LHS
Substitute the identities into the LHS equation: \(1-(1-2\sin^2(\theta))+2\cos^2(2\theta)-1-(1-2\sin^2(3\theta))\).
04
- Simplify by Combining Like Terms
Combine like terms and simplify: \(1-1+2\sin^2(\theta)+2\left(2\cos^2(\theta)-1\right)-1+2\sin^2(3\theta)\) simplifies to \(2\sin^2(\theta)+4\cos^2(\theta)-2+2\sin^2(3\theta)\).
05
- Apply Product-to-Sum Formulas
Use product-to-sum formulas to convert \(4 \ \cos(2\theta) \sin(3\theta) = 4 (\frac{1}{2})(\sin(5\theta) - \sin(\theta)) = 2 (\sin(5\theta) - \sin(\theta))\) Simplify the result to match the RHS:
06
- Establish Relationship
Upon matching the simplified LHS with the RHS: \((1 - \cos (2 \theta) + \cos (4 \theta) - \cos (6 \theta) = 4 \ sin \theta \ cos (2 \theta) \ sin (3 \theta))\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Double Angle Formulas
Double angle formulas help simplify expressions involving trigonometric functions of angles that are double another angle, like how the given exercise involves terms such as \(2\theta\), \(4\theta\), and \(6\theta\). These formulas are derived from sum identities and can be very useful.
Key double angle formulas you'll find handy include:
By substituting these formulas, we can transform intricate trigonometric expressions into something more manageable. This is crucial when you want to combine and simplify terms, as demonstrated in the step-by-step solution.
Key double angle formulas you'll find handy include:
- \(\cos(2\theta) = 1 - 2\sin^2(\theta)\)
- \(\cos(4\theta) = 2\cos^2(2\theta) - 1\)
- \(\cos(6\theta) = 1 - 2\sin^2(3\theta)\)
By substituting these formulas, we can transform intricate trigonometric expressions into something more manageable. This is crucial when you want to combine and simplify terms, as demonstrated in the step-by-step solution.
Product-to-Sum Formulas
Product-to-sum formulas are used to transform products of trigonometric functions into sums or differences. This can be incredibly helpful for simplification.
The key product-to-sum formulas we use are:
In the exercise, transforming \(4 \cos(2\theta) \sin(3\theta)\) using product-to-sum formulas is an essential step that helps to match the Left-Hand Side (LHS) with the Right-Hand Side (RHS). This transformation allows for easier comparison and simplification.
The key product-to-sum formulas we use are:
- \(\cos A \cos B = \frac{1}{2} [\cos(A+B) + \cos(A-B)]\)
- \(\sin A \sin B = \frac{1}{2} [\cos(A-B) - \cos(A+B)]\)
- \(\sin A \cos B = \frac{1}{2} [\sin(A+B) + \sin(A-B)]\)
In the exercise, transforming \(4 \cos(2\theta) \sin(3\theta)\) using product-to-sum formulas is an essential step that helps to match the Left-Hand Side (LHS) with the Right-Hand Side (RHS). This transformation allows for easier comparison and simplification.
Simplifying Trigonometric Expressions
Simplifying trigonometric expressions often requires a combination of methods, including substitution of identities and combining like terms.
The process typically involves:
For example, in step 4 of the solution, substituting double angle identities simplifies the LHS expression. Combining like terms results in the expression \(2\sin^2(\theta) + 4\cos^2(\theta) - 2 + 2\sin^2(3\theta)\), which brings it closer to the form needed to match the RHS.
The process typically involves:
- Identifying the relevant trigonometric identities.
- Substituting these identities into the original expression.
- Combining like terms and performing algebraic manipulations.
For example, in step 4 of the solution, substituting double angle identities simplifies the LHS expression. Combining like terms results in the expression \(2\sin^2(\theta) + 4\cos^2(\theta) - 2 + 2\sin^2(3\theta)\), which brings it closer to the form needed to match the RHS.
Establishing Trigonometric Identities
Establishing or proving trigonometric identities involves demonstrating that two different-looking expressions are indeed equivalent. This is often done through a series of transformations and simplifications.
Here's a quick guideline:
In this exercise, we verified the identity \(1 - \cos (2 \theta) + \cos (4 \theta) - \cos (6 \theta) = 4 \sin \theta \cos (2 \theta) \sin (3 \theta)\) by simplifying the LHS using double angle and product-to-sum formulas and then matching it with the RHS. Each step methodically shows that both sides of the equation are equivalent, thus establishing the identity.
Here's a quick guideline:
- Start by writing both sides of the equation.
- Use known identities to simplify one or both sides.
- Transform these expressions step-by-step until they match.
In this exercise, we verified the identity \(1 - \cos (2 \theta) + \cos (4 \theta) - \cos (6 \theta) = 4 \sin \theta \cos (2 \theta) \sin (3 \theta)\) by simplifying the LHS using double angle and product-to-sum formulas and then matching it with the RHS. Each step methodically shows that both sides of the equation are equivalent, thus establishing the identity.
