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.

검증된 단계별 안내

Identify the given function: \(y = -2 - \cot\left(x - \frac{\pi}{4}\right)\). We want to find the x-intercepts, where \(y = 0\).

Set the function equal to zero to find the x-intercepts: \(0 = -2 - \cot\left(x - \frac{\pi}{4}\right)\).

Rearrange the equation to isolate the cotangent term: \(\cot\left(x - \frac{\pi}{4}\right) = -2\).

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.

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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.