Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
6. b. arccsc(-2/√3)

검증된 단계별 안내

Recall that \( \arccsc(x) \) is the inverse cosecant function, which gives an angle \( \theta \) such that \( \csc(\theta) = x \). Here, we want to find \( \theta = \arccsc\left(-\frac{2}{\sqrt{3}}\right) \).

Rewrite the cosecant in terms of sine: since \( \csc(\theta) = \frac{1}{\sin(\theta)} \), we have \( \sin(\theta) = \frac{1}{\csc(\theta)} = \frac{1}{-\frac{2}{\sqrt{3}}} = -\frac{\sqrt{3}}{2} \).

Determine the reference angle by considering the positive value of sine: \( \sin(\theta_{ref}) = \frac{\sqrt{3}}{2} \). From the unit circle, this corresponds to an angle of \( \frac{\pi}{3} \) radians (or 60 degrees).

Since the sine value is negative and cosecant is negative, identify the quadrant where sine is negative. Sine is negative in Quadrants III and IV. The principal range of \( \arccsc(x) \) is \( [-\frac{\pi}{2}, 0) \cup (0, \frac{\pi}{2}] \) excluding zero, so the angle must be in Quadrant IV (negative angle between 0 and \( -\frac{\pi}{2} \)).

Therefore, the angle \( \theta \) is the negative of the reference angle, so \( \theta = -\frac{\pi}{3} \). This is the angle in the appropriate quadrant corresponding to \( \arccsc\left(-\frac{2}{\sqrt{3}}\right) \).

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주요 개념

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Inverse Trigonometric Functions

Inverse trigonometric functions, like arccsc, return the angle whose trigonometric ratio equals a given value. For arccsc(x), it gives the angle whose cosecant is x. Understanding their domains and ranges is essential to find correct angle values.

Reference Triangles

Reference triangles are right triangles used to relate trigonometric ratios to angles in different quadrants. By constructing a triangle with known side ratios, you can determine the reference angle and then adjust for the correct quadrant.

Quadrant Sign Rules

Trigonometric functions have specific signs in each quadrant. Since arccsc(-2/√3) is negative, the angle lies where cosecant is negative (quadrants II or IV). Knowing these sign rules helps identify the correct quadrant for the angle.

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.6. b. arccsc(-2/√3)