evaluate. write your answer in simplified, rationalized form. do not round. sin 45° =

Explanation:

Step1: Recall sine - value of special angle

In a right - isosceles triangle with angles 45°, 45°, and 90°, if the legs have length (a), by the Pythagorean theorem, the hypotenuse (c=\sqrt{a^{2}+a^{2}}=\sqrt{2a^{2}}=\sqrt{2}a). The sine of an angle in a right - triangle is defined as (\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}). For (\theta = 45^{\circ}), the opposite side to the 45° angle has length (a) and the hypotenuse has length (\sqrt{2}a). So, (\sin45^{\circ}=\frac{a}{\sqrt{2}a}).

Step2: Simplify the fraction

Cancel out the common factor (a) in the numerator and the denominator. Then rationalize the denominator by multiplying the numerator and denominator by (\sqrt{2}). (\frac{a}{\sqrt{2}a}=\frac{1}{\sqrt{2}}=\frac{1\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{\sqrt{2}}{2}).

Answer:

(\frac{\sqrt{2}}{2})