To determine [tex]\(\sin 23^{\circ}\)[/tex], follow these step-by-step instructions:
1. Convert Degrees to Radians: The angle provided is in degrees, and it's common practice to convert this to radians since trigonometric functions like sine are often computed in radians. To convert degrees to radians, utilize the conversion formula:
[tex]\[
\text{Radians} = \text{Degrees} \times \left(\frac{\pi}{180}\right)
\][/tex]
For [tex]\(23^\circ\)[/tex]:
[tex]\[
23 \times \left(\frac{\pi}{180}\right) \approx 0.4014257279586958 \text{ radians}
\][/tex]
2. Calculate the Sine of the Angle in Radians: Now, compute the sine value for the radian measure obtained. The calculation yields:
[tex]\[
\sin(0.4014257279586958) \approx 0.39073112848927377
\][/tex]
3. Match the Calculated Sine Value with Given Options: Compare the calculated value of [tex]\(\sin 23^\circ \approx 0.3907\)[/tex] with the given options:
- A. [tex]\(\frac{5}{13} \approx 0.3846\)[/tex]
- B. [tex]\(\frac{12}{13} \approx 0.9231\)[/tex]
- C. [tex]\(\frac{12}{5} = 2.4\)[/tex]
- D. [tex]\(\frac{5}{12} \approx 0.4167\)[/tex]
Upon comparison, none of the given options A, B, C, or D exactly matches the calculated [tex]\(\sin 23^\circ \approx 0.39073112848927377\)[/tex].
Therefore, the correct value of [tex]\(\sin 23^\circ\)[/tex] is approximately 0.39073112848927377, which does not correspond to any of the provided options.
