Inverse sines and cosines Evaluate or simplify the following expressions without using a calculator.


cos (cos⁻¹ ( -1 ))

Verified step by step guidance

1. Recognize that \( \cos^{-1}(-1) \) is asking for the angle whose cosine is \(-1\).
2. Recall that the range of \( \cos^{-1}(x) \) is \([0, \pi]\).
3. Determine the angle within this range where the cosine value is \(-1\).
4. The angle that satisfies \( \cos(\theta) = -1 \) within the range \([0, \pi]\) is \( \pi \).
5. Substitute \( \pi \) back into the original expression: \( \cos(\pi) \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹(x) and cos⁻¹(x), are used to find angles when given a trigonometric ratio. For example, cos⁻¹(-1) gives the angle whose cosine is -1, which is π radians (or 180 degrees). Understanding these functions is crucial for evaluating expressions involving them.

Cosine Function

The cosine function, denoted as cos(x), relates the angle x in 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 values of cosine at key angles (like 0, π/2, π, etc.) is essential for simplifying expressions involving cosine.

Composition of Functions

Composition of functions involves applying one function to the result of another. In this case, evaluating cos(cos⁻¹(-1)) means finding the cosine of the angle whose cosine is -1. This concept is fundamental in calculus and algebra, as it allows for the simplification of complex expressions by breaking them down into manageable parts.