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Express the following as single trigonometry ratio: \( 3.11 \cos 2 x \cdot \cos 3 x-\sin 2 x \cdot \sin 3 x \)

Ask by Ramirez Long. in South Africa
Feb 27,2025

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Answer

The expression simplifies to \(\frac{411\cos(5x) + 211\cos(x)}{200}\).

Solution

Calculate or simplify the expression \( 3.11 \cos(2x) \cdot \cos(3x)-\sin(2x) \cdot \sin(3x) \). Simplify the expression by following steps: - step0: Solution: \(3.11\cos\left(2x\right)\cos\left(3x\right)-\sin\left(2x\right)\sin\left(3x\right)\) - step1: Transform the expression: \(\frac{311}{100}\cos\left(3x\right)\cos\left(2x\right)-\sin\left(3x\right)\sin\left(2x\right)\) - step2: Transform the expression: \(\frac{311}{100}\cos\left(3x\right)\left(\cos^{2}\left(x\right)-\sin^{2}\left(x\right)\right)-\sin\left(3x\right)\sin\left(2x\right)\) - step3: Transform the expression: \(\frac{311}{100}\cos\left(3x\right)\left(\cos^{2}\left(x\right)-\sin^{2}\left(x\right)\right)-2\sin\left(3x\right)\cos\left(x\right)\sin\left(x\right)\) - step4: Transform the expression: \(\frac{311}{100}\cos\left(3x\right)\left(1-2\sin^{2}\left(x\right)\right)-2\sin\left(3x\right)\cos\left(x\right)\sin\left(x\right)\) - step5: Transform the expression: \(\frac{311}{100}\cos\left(3x\right)-\frac{311}{50}\sin^{2}\left(x\right)\cos\left(3x\right)-2\sin\left(x\right)\sin\left(3x\right)\cos\left(x\right)\) - step6: Transform the expression: \(\frac{311}{100}\cos\left(3x\right)-\frac{311}{50}\cos\left(3x\right)\left(1-\cos^{2}\left(x\right)\right)-2\sin\left(x\right)\sin\left(3x\right)\cos\left(x\right)\) - step7: Transform the expression: \(\frac{311}{100}\cos\left(3x\right)-\frac{311\cos\left(3x\right)\left(1-\cos\left(2x\right)\right)}{100}-2\sin\left(x\right)\sin\left(3x\right)\cos\left(x\right)\) - step8: Subtract the terms: \(\frac{311\cos\left(2x\right)\cos\left(3x\right)}{100}-2\sin\left(x\right)\sin\left(3x\right)\cos\left(x\right)\) - step9: Reduce fractions to a common denominator: \(\frac{311\cos\left(2x\right)\cos\left(3x\right)}{100}-\frac{2\sin\left(x\right)\sin\left(3x\right)\cos\left(x\right)\times 100}{100}\) - step10: Transform the expression: \(\frac{311\cos\left(2x\right)\cos\left(3x\right)-2\sin\left(x\right)\sin\left(3x\right)\cos\left(x\right)\times 100}{100}\) - step11: Multiply the terms: \(\frac{311\cos\left(2x\right)\cos\left(3x\right)-200\sin\left(x\right)\sin\left(3x\right)\cos\left(x\right)}{100}\) - step12: Transform the expression: \(\frac{\frac{411\cos\left(5x\right)+211\cos\left(x\right)}{2}}{100}\) - step13: Multiply by the reciprocal: \(\frac{411\cos\left(5x\right)+211\cos\left(x\right)}{2}\times \frac{1}{100}\) - step14: Multiply the terms: \(\frac{411\cos\left(5x\right)+211\cos\left(x\right)}{2\times 100}\) - step15: Multiply the terms: \(\frac{411\cos\left(5x\right)+211\cos\left(x\right)}{200}\) The expression \(3.11 \cos 2x \cdot \cos 3x - \sin 2x \cdot \sin 3x\) can be simplified to \(\frac{411\cos(5x) + 211\cos(x)}{200}\).

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To express \( 3.11 \cos 2x \cdot \cos 3x - \sin 2x \cdot \sin 3x \) as a single trigonometric ratio, we can make use of the cosine angle addition formula, which states that: \[ \cos(a + b) = \cos a \cos b - \sin a \sin b \] From this formula, we can rewrite the expression as follows: \[ 3.11 \cos 2x \cdot \cos 3x - \sin 2x \cdot \sin 3x = \cos(2x + 3x) = \cos(5x) \] Now, we can factor out the 3.11 from the first part of the expression: \[ = 3.11 \left(\cos 2x \cdot \cos 3x - \frac{1}{3.11} \sin 2x \cdot \sin 3x\right) \] However, this complicates things instead of simplifying them. Thus, the best approach is to recognize that we can still represent the full expression as a linear combination of cosine ratios. Therefore, the best we can directly express here is: \[ 3.11 \cos(2x + 3x) = 3.11 \cos(5x) \] In conclusion, the given expression simplifies to: \[ 3.11 \cos(5x) \] as a single trigonometric ratio.

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