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Solve the equation for \( x \). (Round your answer to three decimal places.) \[ \arctan (2 x-9)=-1 \]

Ask by Curry Salazar. in the United States
Feb 26,2025

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\( x \approx 3.721 \)

Solution

Solve the equation \( \arctan(2x-9)=-1 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\arctan\left(2x-9\right)=-1\) - step1: Use the inverse trigonometric function: \(2x-9=\tan\left(-1\right)\) - step2: Calculate the trigonometric value: \(2x-9=-\tan\left(1\right)\) - step3: Move the constant to the right side: \(2x=-\tan\left(1\right)+9\) - step4: Divide both sides: \(\frac{2x}{2}=\frac{-\tan\left(1\right)+9}{2}\) - step5: Divide the numbers: \(x=\frac{-\tan\left(1\right)+9}{2}\) The solution to the equation \( \arctan(2x-9)=-1 \) is \( x \approx 3.721296 \) when rounded to three decimal places.

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To solve the equation \( \arctan(2x - 9) = -1 \), we start by taking the tangent on both sides to eliminate the arctangent: \[ 2x - 9 = \tan(-1) \] We know that \( \tan(-1) \) is equal to \(-\tan(1)\). The value of \(\tan(1)\) is approximately \(1.5574\), thus: \[ \tan(-1) \approx -1.5574 \] Plugging that into our equation gives: \[ 2x - 9 = -1.5574 \] Now, we solve for \(x\): \[ 2x = -1.5574 + 9 \] \[ 2x = 7.4426 \] \[ x = \frac{7.4426}{2} \] \[ x \approx 3.7213 \] Rounding this to three decimal places, we get: \[ \boxed{3.721} \]

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