Solve each equation for x.
arccos x + arctan 1 = 11π/12

검증된 단계별 안내

Recognize that the equation is given as \(\arccos x + \arctan 1 = \frac{11\pi}{12}\), and our goal is to solve for \(x\).

Recall the value of \(\arctan 1\). Since \(\tan \frac{\pi}{4} = 1\), it follows that \(\arctan 1 = \frac{\pi}{4}\).

Substitute \(\arctan 1 = \frac{\pi}{4}\) into the equation to get \(\arccos x + \frac{\pi}{4} = \frac{11\pi}{12}\).

Isolate \(\arccos x\) by subtracting \(\frac{\pi}{4}\) from both sides: \(\arccos x = \frac{11\pi}{12} - \frac{\pi}{4}\).

Simplify the right side by finding a common denominator and then use the definition of arccosine to write \(x = \cos\left(\arccos x\right) = \cos\left(\frac{11\pi}{12} - \frac{\pi}{4}\right)\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arccos and arctan, return the angle whose trigonometric ratio equals a given value. For example, arccos x gives the angle whose cosine is x, and arctan y gives the angle whose tangent is y. Understanding their ranges and outputs is essential for solving equations involving these functions.

Properties and Values of Special Angles

Special angles like π/4, π/3, and π/6 have well-known sine, cosine, and tangent values. Recognizing that arctan 1 equals π/4 helps simplify the equation. Familiarity with these values allows substitution and manipulation of trigonometric expressions to isolate variables.

Solving Trigonometric Equations

Solving equations involving inverse trig functions often requires isolating the inverse function, using known angle values, and applying algebraic manipulation. Understanding how to rewrite the equation and use identities or known values is key to finding the solution for x.

Solve each equation for x.arccos x + arctan 1 = 11π/12