Use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.


<IMAGE>


csc 45°

Key Concepts

Cosecant Function

The cosecant function, denoted as csc, is the reciprocal of the sine function. For an angle θ, csc(θ) = 1/sin(θ). This means that to find csc(45°), one must first determine sin(45°), which is √2/2. Thus, csc(45°) equals 1/(√2/2) = √2.

Special Angles in Trigonometry

Special angles, such as 30°, 45°, and 60°, have known sine, cosine, and tangent values that are commonly used in trigonometric calculations. For example, sin(45°) = √2/2 and cos(45°) = √2/2. Recognizing these values allows for quicker evaluations of trigonometric functions without needing a calculator.

Rationalizing the Denominator

Rationalizing the denominator is a technique used to eliminate square roots or irrational numbers from the denominator of a fraction. This is done by multiplying the numerator and denominator by a suitable value. For instance, to rationalize 1/√2, multiply by √2/√2 to get √2/2, which is a more standard form.