Evaluate cos⁻¹(cos(5π/4)).

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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} $.

The range of $ \cos^{-1}(x) $ is $[0, \pi]$.

Determine the angle $ 5\pi/4 $ in radians, which is outside the range $[0, \pi]$.

Find an equivalent angle within the range $[0, \pi]$ that has the same cosine value as $ 5\pi/4 $.

Use the property $ \cos(\theta) = \cos(2\pi - \theta) $ to find the equivalent angle within the range.

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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.

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Derivatives of Other Inverse Trigonometric Functions

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.

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Graph of Sine and Cosine 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.

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