Give the exact value of each expression. See Example 5. csc 60°

Verified step by step guidance

1. Recall that the cosecant function is the reciprocal of the sine function, so \(\csc \theta = \frac{1}{\sin \theta}\).
2. Identify the angle given: \(60^\circ\).
3. Find the exact value of \(\sin 60^\circ\). From the special angles in trigonometry, \(\sin 60^\circ = \frac{\sqrt{3}}{2}\).
4. Use the reciprocal relationship to write \(\csc 60^\circ = \frac{1}{\sin 60^\circ} = \frac{1}{\frac{\sqrt{3}}{2}}\).
5. Simplify the expression by multiplying numerator and denominator appropriately to find the exact value of \(\csc 60^\circ\).

Key Concepts

Definition of Cosecant (csc)

Cosecant is the reciprocal of the sine function. For any angle θ, csc(θ) = 1/sin(θ). Understanding this relationship allows you to find the cosecant value by first determining the sine of the angle.

Exact Values of Sine for Special Angles

Certain angles like 30°, 45°, and 60° have well-known exact sine values derived from special right triangles. For 60°, sin(60°) = √3/2, which is essential for calculating csc(60°) exactly.

Reciprocal Function Calculation

To find the exact value of csc(60°), you take the reciprocal of sin(60°). This involves flipping the fraction representing sin(60°), turning √3/2 into 2/√3, which can be simplified further if needed.