Find the exact value of each real number y. Do not use a calculator.
y = arccot (―1)

검증된 단계별 안내

Recall the definition of the arccotangent function: \(y = \arccot(x)\) means \(\cot(y) = x\) and \(y\) lies in the principal range of arccot, which is usually \(0 < y < \pi\) for real numbers.

Set up the equation from the problem: \(\cot(y) = -1\).

Recall that \(\cot(y) = \frac{\cos(y)}{\sin(y)}\). So, \(\frac{\cos(y)}{\sin(y)} = -1\) implies \(\cos(y) = -\sin(y)\).

Divide both sides by \(\cos(y)\) (assuming \(\cos(y) \neq 0\)) to get \(1 = -\tan(y)\), or equivalently \(\tan(y) = -1\).

Find the angle \(y\) in the interval \((0, \pi)\) where \(\tan(y) = -1\). This corresponds to the angle where tangent is negative and equals \(-1\).

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주요 개념

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

Inverse Cotangent Function (arccot)

The arccot function is the inverse of the cotangent function, returning an angle whose cotangent is the given value. It is important to understand its principal value range, typically (0, π), to find the correct angle without ambiguity.

Cotangent Function and Its Values

Cotangent is defined as the ratio of the adjacent side to the opposite side in a right triangle, or cot(θ) = cos(θ)/sin(θ). Knowing common cotangent values, such as cot(3π/4) = -1, helps in identifying the exact angle corresponding to a given cotangent value.

Exact Values of Special Angles

Certain angles like π/4, π/3, and π/6 have well-known trigonometric values. Recognizing these special angles and their cotangent values allows for determining exact values without a calculator, which is essential for solving inverse trigonometric problems.

Find the exact value of each real number y. Do not use a calculator.y = arccot (―1)