What is the exact value of $\sin 45°$?

• A. $\frac{\sqrt{2}}{2}$
• B. $\sqrt{3}$
• C. $\frac{1}{2}$
• D. $\frac{1}{\sqrt{2}}$

Answer: (A)

Verified step by step guidance

1. Recognize that the problem asks for the exact value of \(\sin 45^\circ\), which is a common special angle in trigonometry.
2. Recall that \$45^\circ$ is one of the angles in a right isosceles triangle, where the two legs are equal in length.
3. Use the fact that in a right isosceles triangle, the hypotenuse is \(\sqrt{2}\) times the length of each leg, so if each leg is 1, the hypotenuse is \(\sqrt{2}\).
4. Apply the definition of sine as the ratio of the length of the side opposite the angle to the hypotenuse: \(\sin 45^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}}\).
5. Rationalize the denominator by multiplying numerator and denominator by \(\sqrt{2}\) to get \(\sin 45^\circ = \frac{\sqrt{2}}{2}\), which is the exact value.