Evaluate each expression without using a calculator.
cos (csc⁻¹ (-2))

Guida verificata passo dopo passo

Recognize that the expression involves the inverse cosecant function: \(\csc^{-1}(-2)\). Let \(\theta = \csc^{-1}(-2)\), which means \(\csc \theta = -2\).

Recall the definition of cosecant: \(\csc \theta = \frac{1}{\sin \theta}\). From \(\csc \theta = -2\), we get \(\sin \theta = \frac{1}{-2} = -\frac{1}{2}\).

Determine the quadrant where \(\theta\) lies. Since \(\csc^{-1} x\) typically returns values in \([-\frac{\pi}{2}, 0) \cup (0, \frac{\pi}{2}]\) excluding zero, and \(\sin \theta = -\frac{1}{2}\) is negative, \(\theta\) must be in the fourth quadrant.

Use the Pythagorean identity to find \(\cos \theta\): \(\cos \theta = \pm \sqrt{1 - \sin^2 \theta} = \pm \sqrt{1 - \left(-\frac{1}{2}\right)^2} = \pm \sqrt{1 - \frac{1}{4}} = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2}\).

Determine the sign of \(\cos \theta\) in the fourth quadrant. Since cosine is positive in the fourth quadrant, \(\cos \theta = \frac{\sqrt{3}}{2}\). Therefore, \(\cos(\csc^{-1}(-2)) = \frac{\sqrt{3}}{2}\).

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