Consider the following function from Example 5. Work these exercises in order.
y = -2 - cot (x - π/4)
Use the fact that the period of this function is π to find the next positive x-intercept. Round to the nearest hundredth.

Verified step by step guidance

1. Identify the given function: \(y = -2 - \cot\left(x - \frac{\pi}{4}\right)\). We want to find the x-intercepts, where \(y = 0\).
2. Set the function equal to zero to find the x-intercepts: \(0 = -2 - \cot\left(x - \frac{\pi}{4}\right)\).
3. Rearrange the equation to isolate the cotangent term: \(\cot\left(x - \frac{\pi}{4}\right) = -2\).
4. Recall that the cotangent function has a period of \(\pi\), so the general solution for \(\cot \theta = -2\) is \(\theta = \cot^{-1}(-2) + k\pi\), where \(k\) is any integer.
5. Substitute back \(\theta = x - \frac{\pi}{4}\) and solve for \(x\): \(x = \cot^{-1}(-2) + \frac{\pi}{4} + k\pi\). Use the smallest positive \(x\)-intercept found (for some integer \(k\)) and then add the period \(\pi\) to find the next positive x-intercept. Round your answer to the nearest hundredth.

Key Concepts

Cotangent Function and Its Properties

The cotangent function, cot(x), is the reciprocal of the tangent function and is defined as cos(x)/sin(x). It has vertical asymptotes where sin(x) = 0 and zeros where cos(x) = 0. Understanding its behavior and graph is essential for identifying intercepts and transformations.

Period of Trigonometric Functions

The period of a trigonometric function is the length of one complete cycle before the function repeats. For cotangent, the standard period is π. When the function is transformed, such as cot(x - π/4), the period remains π, which helps in finding subsequent intercepts by adding multiples of the period.

Finding X-Intercepts of Transformed Functions

X-intercepts occur where the function equals zero. For y = -2 - cot(x - π/4), set y = 0 and solve for x. This involves isolating cot(x - π/4) and using the periodicity to find the next positive solution, then rounding the result as required.