Find the Exact Value by Using a Half-Angle Identity $\sin 22.5 ^ { \circ }$

question 303

Multiple Choice

Find the exact value by using a half-angle identity.
- $\sin 22.5 ^ { \circ }$

• A) \(- \frac { 1 } { 2 } \sqrt { 2 + \sqrt { 2 } }\)
• B) \(\frac { 1 } { 2 } \sqrt { 2 + \sqrt { 2 } }\)
• C) \(\frac { 1 } { 2 } \sqrt { 2 - \sqrt { 2 } }\)
• D) \(- \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 2 } }\)

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Definitions:

Carbonyl Carbon

The carbonyl carbon is the carbon atom of a carbonyl group (C=O), doubly bonded to an oxygen atom and bonded to two other atoms or groups, which can vary depending on the specific carbonyl compound (e.g., aldehydes, ketones).

Cyclohexylacetic Acid

An organic compound featuring a cyclohexane ring attached to an acetic acid group, used in the synthesis of various pharmaceuticals and chemical products.

Synthetic Route

An outlined series of chemical reactions leading to the synthesis of a complex molecule from simpler starting materials.

Methylenecyclohexane

A cyclic hydrocarbon consisting of a cyclohexane ring with a methylene (-CH2-) group attached to it.