Question

The value of \( \sin 22^{\frac{1}{2}} \) is:

• A. \( \frac{\sqrt{2} - 1}{2\sqrt{2}} \)
• B. \( \frac{\sqrt{2} - 1}{2} \)
• C. \( \frac{\sqrt{2} + 1}{2\sqrt{2}} \)
• D. None of these

Hint:

For exact trigonometric values of uncommon angles like \( 22^{\frac{1}{2}} \), using trigonometric tables or a scientific calculator is the most effective method.

Answer 1

The Correct Option is A

Step 1: Express \( 22^\circ 30' \) in decimal form.
We know that \( 30' \) (minutes) is \( \frac{30}{60} = 0.5 \) degrees.
Therefore, \( 22^\circ 30' = 22.5^\circ \). Step 2: Use the half-angle identity for sine.
The half-angle formula for sine is: \[ \sin \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 - \cos \theta}{2}} \] For \( \theta = 45^\circ \), since \( 22.5^\circ = \frac{45^\circ}{2} \), we can use the half-angle identity: \[ \sin 22.5^\circ = \sqrt{\frac{1 - \cos 45^\circ}{2}} \] Step 3: Find \( \cos 45^\circ \).
We know that: \[ \cos 45^\circ = \frac{\sqrt{2}}{2} \] Step 4: Substitute \( \cos 45^\circ = \frac{\sqrt{2}}{2} \) into the half-angle formula.
\[ \sin 22.5^\circ = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} \] Step 5: Simplify the expression.
First, simplify the expression inside the square root: \[ 1 - \frac{\sqrt{2}}{2} = \frac{2}{2} - \frac{\sqrt{2}}{2} = \frac{2 - \sqrt{2}}{2} \] Now, divide by 2: \[ \frac{2 - \sqrt{2}}{4} \] Thus, the expression becomes: \[ \sin 22.5^\circ = \sqrt{\frac{2 - \sqrt{2}}{4}} \] Step 6: Rationalize the expression.
The final expression for \( \sin 22.5^\circ \) simplifies to: \[ \sin 22.5^\circ = \frac{\sqrt{2} - 1}{2\sqrt{2}} \] Thus, the value of \( \sin 22.5^\circ \) is \( \frac{\sqrt{2} - 1}{2\sqrt{2}} \).