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


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csc 45°

Verified step by step guidance

1. Recall the definition of cosecant: \(\csc \theta = \frac{1}{\sin \theta}\).
2. Identify the value of \(\sin 45^\circ\). Since \(45^\circ\) is a special angle, \(\sin 45^\circ = \frac{\sqrt{2}}{2}\).
3. Substitute \(\sin 45^\circ\) into the cosecant formula: \(\csc 45^\circ = \frac{1}{\frac{\sqrt{2}}{2}}\).
4. Simplify the fraction by multiplying numerator and denominator appropriately: \(\csc 45^\circ = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}}\).
5. Rationalize the denominator by multiplying numerator and denominator by \(\sqrt{2}\): \(\csc 45^\circ = \frac{2}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\sqrt{2}}{2}\), then simplify the fraction.

Key Concepts

Definition of Cosecant (csc)

Cosecant is the reciprocal of sine, defined as csc θ = 1/sin θ. For an angle in a right triangle, it equals the ratio of the hypotenuse to the opposite side. Understanding this helps in evaluating csc 45° using known sine values.

Special Angles and Their Trigonometric Values

Angles like 45° have well-known sine and cosine values derived from special right triangles (e.g., isosceles right triangle). For 45°, sin 45° = √2/2, which simplifies calculations of trigonometric functions such as cosecant.

Rationalizing the Denominator

Rationalizing the denominator involves eliminating square roots from the denominator of a fraction by multiplying numerator and denominator by a suitable radical. This process simplifies expressions and is often required for final answers in trigonometry.