Proving the Trigonometric Identity: Debunking 3sin(a) 4cos(b) 5sin(a b)

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When discussing trigonometric identities, it's essential to understand which identities hold under various transformations. One such commonly misunderstood identity is 3sin(a) 4cos(b) 5sin(a b). This expression does not represent a validated trigonometric identity for all values of a and b. Let's delve into the reasons why and explore the correct formulas and methods for proving trigonometric identities.

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Understanding Trigonometric Identities

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Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. These identities often simplify expressions, solve equations, and are fundamental in various fields of mathematics and physics. One of the most well-known trigonometric identities is the addition formula for sine:

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sin(a b) sin(a)cos(b) sin(b)cos(a)

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This identity is crucial for proving and manipulating expressions involving trigonometric functions. However, proving or disproving other identities requires a careful approach.

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Debunking the Myth: 3sin(a) 4cos(b) 5sin(a b)

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The identity 3sin(a) 4cos(b) 5sin(a b) does not hold true for all values of a and b. To disprove this, let's consider a counterexample. Choose simple values for a and b, such as a π/2 and b 0. Substituting these values, we get:

r r r 3sin(π/2) 4cos(0) 3 * 1 4 * 1 7r 5sin(π/2 0) 5sin(π/2) 5 * 1 5r r r

Clearly, 7 ≠ 5, which demonstrates that the given identity is incorrect.

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Proving Trigonometric Identities Correctly

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To correctly prove trigonometric identities, one must use established formulas and algebraic manipulations. Here, we'll use the correct addition formula for sine to illustrate the proper approach:

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Sin(a b) Proof

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Start with the known formula:

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sin(a b) sin(a)cos(b) sin(b)cos(a)

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This formula can be derived using the unit circle and geometric constructions, but for our purposes, we will use it directly.

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Proving a Different Identity as an Example

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Let's prove a different, correct identity: sin(a b) sin(a)cos(b) sin(b)cos(a)

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Proof:

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Step 1: Start with the Addition Formula

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Using the given addition formula for sine, we have:

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sin(a b) sin(a)cos(b) sin(b)cos(a)

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Step 2: Verify Through Algebraic Manipulation

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To verify this identity, we can use the definitions of sine and cosine in terms of the unit circle or Euler's formula. However, for simplicity, we will use the known formula and properties of trigonometric functions.

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Let's assume that the values of a and b are arbitrary. We need to show that the formula holds for any a and b. We can do this by:

r r r Evaluating both sides of the equation for specific values of a and b to verify consistency.r Using geometric or algebraic proofs to show that the formula is valid for all values of a and b.r r r

Step 3: General Verification

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Consider the unit circle or complex plane representations of sine and cosine. The addition formula is derived from the geometric properties of these functions. For example, in the complex plane, we can express sine and cosine in terms of exponentials using Euler's formula:

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e^(ix) cos(x) i sin(x)

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Multiplying these formulas and comparing the real and imaginary parts can provide additional proof.

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Conclusion and Final Thoughts

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In conclusion, the identity 3sin(a) 4cos(b) 5sin(a b) is incorrect and does not hold for all values of a and b. It is crucial to understand the correct trigonometric identities and the methods for proving them. Using the proven addition formula for sine, we can verify that sin(a b) sin(a)cos(b) sin(b)cos(a) is a valid identity.

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For accurate and reliable information in trigonometry, always refer to well-established formulas and proofs. Understanding and using these identities can significantly enhance your problem-solving skills in mathematics and related fields.

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