Find the exact value of each real number y if it exists. Do not use a calculator.
y = cos⁻¹ (―1)

검증된 단계별 안내

Recall that the function \( \cos^{-1}(x) \), also known as arccosine, gives the angle \( y \) in the range \( 0 \leq y \leq \pi \) such that \( \cos(y) = x \).

Identify the value inside the arccosine function: here, \( x = -1 \). So we need to find \( y \) such that \( \cos(y) = -1 \).

Think about the unit circle and the cosine values at key angles between \( 0 \) and \( \pi \). Cosine corresponds to the x-coordinate on the unit circle.

Recall that \( \cos(\pi) = -1 \), and since \( \pi \) is within the principal range of arccosine, this is the angle we are looking for.

Therefore, the exact value of \( y = \cos^{-1}(-1) \) is \( y = \pi \).

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주요 개념

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

Inverse Cosine Function (cos⁻¹ or arccos)

The inverse cosine function, denoted as cos⁻¹ or arccos, returns the angle whose cosine value is a given number. It is defined for inputs between -1 and 1 and outputs angles in the range [0, π] radians (or [0°, 180°]). Understanding this range is crucial for finding exact angle values.

Cosine Values of Special Angles

Certain angles have well-known cosine values, such as cos(0) = 1, cos(π/2) = 0, and cos(π) = -1. Recognizing these special angles helps in determining the exact value of inverse trigonometric functions without a calculator.

Exact Values vs. Approximate Values

In trigonometry, exact values are expressed in terms of well-known constants like π or simple fractions, rather than decimal approximations. For inverse cosine, this means identifying the angle in radians or degrees that precisely corresponds to the given cosine value.

Find the exact value of each real number y if it exists. Do not use a calculator.y = cos⁻¹ (―1)