Question

express sin e as a fraction in simplest terms.
answer attempt 1 out of 2
\\( \sin e = \\)
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Explanation:

Step1: Recall the definition of sine in a right - triangle

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). For \(\angle E\), the side opposite to \(\angle E\) is \(CD\) and the hypotenuse is \(CE\).

Step2: Identify the lengths of the sides

We are given that \(CD = 2\) and \(CE=\sqrt{CD^{2}+DE^{2}}\) (by Pythagoras' theorem \(CE=\sqrt{2^{2}+9^{2}}=\sqrt{4 + 81}=\sqrt{85}\)), but using the sine formula directly \(\sin E=\frac{CD}{CE}\). Since \(CD = 2\) and \(CE\) is the hypotenuse of right - triangle \(CDE\) with legs \(CD = 2\) and \(DE=9\), by the definition of sine in a right - triangle \(\sin E=\frac{CD}{CE}\) (where \(CE=\sqrt{CD^{2}+DE^{2}}\)), but more simply, using the formula \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\) for right - triangle \(CDE\) with right - angle at \(D\), \(\sin E=\frac{CD}{\sqrt{CD^{2}+DE^{2}}}\). Substituting \(CD = 2\) and \(DE = 9\), we get \(\sin E=\frac{2}{\sqrt{4 + 81}}=\frac{2}{\sqrt{85}}\). Rationalizing the denominator (multiplying numerator and denominator by \(\sqrt{85}\)), we have \(\sin E=\frac{2\sqrt{85}}{85}\).

Another way: Using the formula \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\) directly from the right - triangle \(CDE\) (right - angled at \(D\)), \(\sin E=\frac{CD}{CE}\). Since \(CD = 2\) and \(CE=\sqrt{2^{2}+9^{2}}=\sqrt{85}\), \(\sin E=\frac{2}{\sqrt{85}}\).

Answer:

\(\frac{2}{\sqrt{85}}\) (or \(\frac{2\sqrt{85}}{85}\) after rationalizing)