Question

Solve the equation for \(x\). $$ \arctan (2 x-5)=-1 $$

Short answer

The solution for the equation is \(x = 2\).

Step-by-step solution

- Notice Operand

Notice the term \(-1\) on the right side of the equation, this indicates that \(\arctan(2x-5)\) is equal to an angle whose tangent value is \(-1\). From the knowledge of trigonometry, we know that \(\tan(-\pi/4) = -1\). So we can set \(-\pi/4 = \arctan(2x-5)\), since the solution lies within the interval \(-\pi/2 \leq x \leq \pi/2\).

- Apply Tangent

Apply tangent to both sides, which gives us \(\tan(-\pi/4) = \tan (\arctan (2x-5))\). This simplifies to \(-1 = 2x - 5\).

- Isolate for x

Rearrange to isolate \(x\). Add 5 to both sides: \(5 - 1 = 2x\). Then divide both sides by 2: \((5 - 1)/2 = x\).