CONCEPT PREVIEW Match each trigonometric function in Column I with its value in Column II. Choices may be used once, more than once, or not at all.
csc 60°

Verified step by step guidance

1. Step 1: Recall the definition of the cosecant function. Cosecant is the reciprocal of sine, so \(\csc \theta = \frac{1}{\sin \theta}\).
2. Step 2: Find the value of \(\sin 60^\circ\). From the special angles in trigonometry, \(\sin 60^\circ = \frac{\sqrt{3}}{2}\).
3. Step 3: Calculate \(\csc 60^\circ\) using the reciprocal relationship: \(\csc 60^\circ = \frac{1}{\sin 60^\circ} = \frac{1}{\frac{\sqrt{3}}{2}}\).
4. Step 4: Simplify the expression for \(\csc 60^\circ\) by multiplying numerator and denominator appropriately: \(\csc 60^\circ = \frac{2}{\sqrt{3}}\).
5. Step 5: Rationalize the denominator if needed by multiplying numerator and denominator by \(\sqrt{3}\) to get \(\csc 60^\circ = \frac{2\sqrt{3}}{3}\), which matches one of the values in Column II.

Key Concepts

Basic Trigonometric Ratios

Trigonometric functions such as sine, cosine, and cosecant relate the angles of a right triangle to the ratios of its sides. For example, csc θ is the reciprocal of sin θ, meaning csc θ = 1/sin θ. Understanding these ratios is essential for matching functions to their numerical values.

Special Angles and Their Values

Certain angles like 30°, 45°, and 60° have well-known exact trigonometric values involving square roots and fractions. For instance, sin 60° = √3/2 and csc 60° = 2/√3. Memorizing these special angle values helps quickly identify correct matches in problems.

Reciprocal Identities

Reciprocal identities express functions like cosecant, secant, and cotangent as reciprocals of sine, cosine, and tangent respectively. For example, csc θ = 1/sin θ. Recognizing these identities allows conversion between functions and their values, aiding in matching tasks.