Evaluate cos⁻¹(cos(5π/4)).

Verified step by step guidance

1. Recognize that \( \cos^{-1}(x) \) is the inverse function of \( \cos(x) \), which means \( \cos^{-1}(\cos(x)) = x \) for \( x \) in the range of \( \cos^{-1} \).
2. The range of \( \cos^{-1}(x) \) is \([0, \pi]\).
3. Determine the angle \( 5\pi/4 \) in radians, which is outside the range \([0, \pi]\).
4. Find an equivalent angle within the range \([0, \pi]\) that has the same cosine value as \( 5\pi/4 \).
5. Use the property \( \cos(\theta) = \cos(2\pi - \theta) \) to find the equivalent angle within the range.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as cos⁻¹(x), are used to find the angle whose cosine is x. These functions have specific ranges; for cos⁻¹(x), the output is restricted to the interval [0, π]. This means that when evaluating cos⁻¹(cos(θ)), the result will depend on the angle θ and its position within the defined range.

Cosine Function and Its Periodicity

The cosine function is periodic with a period of 2π, meaning that cos(θ) = cos(θ + 2kπ) for any integer k. This periodicity implies that angles can be expressed in multiple equivalent forms. For example, cos(5π/4) can be simplified by recognizing that it is equivalent to cos(5π/4 - 2π) = cos(-3π/4), which helps in evaluating the inverse function.

Quadrants and Reference Angles

Understanding the unit circle and the quadrants is essential for evaluating trigonometric functions. The angle 5π/4 is located in the third quadrant, where cosine values are negative. The reference angle for 5π/4 is π/4, which helps in determining the cosine value, as cos(5π/4) = -√2/2. This knowledge is crucial for correctly applying the inverse function.