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