Solve each equation for exact solutions.
4/3 cos⁻¹ x/4 = π

검증된 단계별 안내

Start by isolating the inverse cosine expression. Multiply both sides of the equation by \( \frac{3}{4} \) to get \( \cos^{-1}\left( \frac{x}{4} \right) = \frac{3}{4} \pi \).

Recall that \( \cos^{-1}(y) = \theta \) means \( \cos(\theta) = y \). So rewrite the equation as \( \cos\left( \frac{3}{4} \pi \right) = \frac{x}{4} \).

Evaluate \( \cos\left( \frac{3}{4} \pi \right) \) using the unit circle or known cosine values for special angles. Remember that \( \frac{3}{4} \pi = 135^\circ \) and cosine is negative in the second quadrant.

Set \( \frac{x}{4} \) equal to the value found in step 3, then solve for \( x \) by multiplying both sides by 4.

Consider the domain of \( \cos^{-1} \), which is \( [-1,1] \), to verify that the solution for \( x \) is valid. If necessary, check for any additional solutions within the domain.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

도움이 되었나요?

주요 개념

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

Inverse Cosine Function (cos⁻¹ or arccos)

The inverse cosine function returns the angle whose cosine is a given number. Its output range is from 0 to π radians. Understanding this function is essential to isolate the variable inside the cosine and solve for exact angle values.

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and then applying inverse functions to find the angle. It also requires considering the domain and range of the inverse function to determine all possible solutions.

Manipulating Algebraic Expressions

Algebraic manipulation is necessary to isolate the variable inside the inverse cosine function. This includes multiplying or dividing both sides of the equation and simplifying fractions to express the variable clearly for substitution or evaluation.

Solve each equation for exact solutions.4/3 cos⁻¹ x/4 = π