Find the exact value of each real number y. Do not use a calculator.
y = cos⁻¹ (―√2/2)

검증된 단계별 안내

Recognize that the problem asks for the angle \( y \) such that \( \cos(y) = -\frac{\sqrt{2}}{2} \). This means we need to find the angle whose cosine value is \( -\frac{\sqrt{2}}{2} \).

Recall the unit circle values for cosine. The value \( \frac{\sqrt{2}}{2} \) corresponds to angles \( \frac{\pi}{4} \) and \( \frac{7\pi}{4} \) in the first and fourth quadrants, respectively. Since the cosine is negative, \( y \) must be in the second or third quadrant where cosine values are negative.

Identify the angles in the second and third quadrants where cosine equals \( -\frac{\sqrt{2}}{2} \). These are \( \frac{3\pi}{4} \) (second quadrant) and \( \frac{5\pi}{4} \) (third quadrant).

Recall that the range of the inverse cosine function \( \cos^{-1}(x) \) is \( [0, \pi] \), which means the output angle \( y \) must lie between 0 and \( \pi \).

Therefore, select the angle within the range \( [0, \pi] \) where \( \cos(y) = -\frac{\sqrt{2}}{2} \), which is \( y = \frac{3\pi}{4} \).

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. It is defined for inputs between -1 and 1 and outputs angles in the range [0, π] radians. Understanding this helps find the angle y such that cos(y) equals the given value.

Exact Values of Cosine for Special Angles

Certain angles have well-known cosine values expressed in terms of square roots, such as cos(π/4) = √2/2. Recognizing these exact values allows you to identify the angle corresponding to a given cosine value without a calculator.

Handling Negative Cosine Values

Cosine is negative in the second quadrant (angles between π/2 and π). Since arccos outputs angles in [0, π], a negative cosine value corresponds to an angle in this range, typically π minus the reference angle with positive cosine.

Find the exact value of each real number y. Do not use a calculator.y = cos⁻¹ (―√2/2)