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

Verified step by step guidance

1. Recognize that the expression involves the inverse cosecant function: \(\csc^{-1}(-2)\). Let \(\theta = \csc^{-1}(-2)\), which means \(\csc \theta = -2\).
2. Recall the definition of cosecant: \(\csc \theta = \frac{1}{\sin \theta}\). From \(\csc \theta = -2\), we get \(\sin \theta = \frac{1}{-2} = -\frac{1}{2}\).
3. 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.
4. Use the Pythagorean identity to find \(\cos \theta\): \(\cos \theta = \pm \sqrt{1 - \sin^2 \theta} = \pm \sqrt{1 - \left(-\frac{1}{2}\right)^2} = \pm \sqrt{1 - \frac{1}{4}} = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2}\).
5. 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, \(\cos(\csc^{-1}(-2)) = \frac{\sqrt{3}}{2}\).

Key Concepts

Inverse Cosecant Function (csc⁻¹)

The inverse cosecant function, csc⁻¹(x), returns an angle whose cosecant is x. Since cosecant is the reciprocal of sine, csc⁻¹(x) gives an angle θ such that sin(θ) = 1/x. Understanding the domain and range of csc⁻¹ is essential for correctly interpreting the angle.

Relationship Between Sine and Cosine

Sine and cosine are fundamental trigonometric functions related by the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Knowing one value allows you to find the other, considering the quadrant of the angle to determine the sign of cosine.

Evaluating Trigonometric Expressions Without a Calculator

To evaluate expressions like cos(csc⁻¹(-2)) without a calculator, use known values and identities. Start by finding the angle from the inverse function, then apply trigonometric identities and sign rules based on the angle's quadrant to find the exact value.