Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
4. b. arcsin(-1/√2)

검증된 단계별 안내

Recall that the function \(\arcsin(x)\) gives the angle \(\theta\) whose sine is \(x\), with the range of \(\arcsin\) restricted to \([-\frac{\pi}{2}, \frac{\pi}{2}]\) (Quadrants IV and I).

Identify the value inside the \(\arcsin\): here it is \(-\frac{1}{\sqrt{2}}\). Recognize that \(\sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\), so the reference angle is \(\frac{\pi}{4}\).

Since the sine value is negative, and \(\arcsin\) outputs angles in \([-\frac{\pi}{2}, \frac{\pi}{2}]\), the angle must be in Quadrant IV where sine is negative.

Use the reference triangle in Quadrant IV to express the angle as \(-\frac{\pi}{4}\), because sine of \(-\frac{\pi}{4}\) is \(-\frac{1}{\sqrt{2}}\).

Therefore, the angle \(\theta = \arcsin\left(-\frac{1}{\sqrt{2}}\right)\) corresponds to \(-\frac{\pi}{4}\) within the principal range of \(\arcsin\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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주요 개념

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

Inverse Sine Function (arcsin)

The inverse sine function, arcsin, returns the angle whose sine value is a given number. Its range is limited to angles between -π/2 and π/2 (or -90° to 90°), meaning it outputs angles in the first and fourth quadrants. Understanding this helps identify the principal value of the angle.

Reference Triangles

Reference triangles are right triangles used to find angle measures based on known trigonometric ratios. By considering the absolute value of the sine and the quadrant of the angle, you can determine the exact angle measure and its sign, aiding in solving inverse trig problems.

Quadrants and Sign of Trigonometric Functions

The sign of sine depends on the quadrant where the angle lies: sine is positive in the first and second quadrants and negative in the third and fourth. Knowing the quadrant helps determine the correct angle corresponding to a given sine value, especially when the value is negative.

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.4. b. arcsin(-1/√2)