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

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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