Evaluate each expression without using a calculator.
cos (arccos (-1))

Verified step by step guidance

1. Understand that \( \arccos(x) \) is the inverse function of \( \cos(x) \), which means \( \arccos(x) \) gives the angle whose cosine is \( x \).
2. Recognize that \( \arccos(-1) \) is asking for the angle whose cosine is \(-1\).
3. Recall that the cosine of \( \pi \) (or 180 degrees) is \(-1\).
4. Therefore, \( \arccos(-1) = \pi \).
5. Since \( \cos(\arccos(x)) = x \), substitute \( x = -1 \) to find \( \cos(\arccos(-1)) = -1 \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arccos, are used to find the angle whose cosine is a given value. For example, arccos(-1) returns the angle in the range [0, π] where the cosine equals -1, which is π radians. Understanding these functions is crucial for evaluating expressions involving angles and their trigonometric ratios.

Cosine Function

The cosine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. It is periodic and ranges from -1 to 1. Knowing the properties of the cosine function helps in evaluating expressions like cos(arccos(-1)), as it directly connects the angle to its cosine value.

Composition of Functions

Composition of functions involves applying one function to the result of another. In this case, evaluating cos(arccos(-1)) requires understanding that the output of arccos(-1) is an angle, which is then used as the input for the cosine function. This concept is essential for simplifying expressions and understanding how functions interact.