Question 2 The cosine of an angle in Quadrant II is $-\frac{2}{3}$. Find the sine of the angle.
Solution
Step 1 ：The problem states that the cosine of an angle in Quadrant II is \(-\frac{2}{3}\).
Step 2 ：In the second quadrant, the sine of an angle is positive.
Step 3 ：We can use the Pythagorean identity for sine and cosine, which states that \(\sin^2(\theta) + \cos^2(\theta) = 1\).
Step 4 ：We can rearrange this to find \(\sin(\theta)\), given that we know \(\cos(\theta)\).
Step 5 ：Substituting \(\cos(\theta) = -\frac{2}{3}\) into the Pythagorean identity, we get \(\sin^2(\theta) = 1 - \left(-\frac{2}{3}\right)^2\).
Step 6 ：Solving for \(\sin(\theta)\), we get two possible solutions: \(\sin(\theta) = \sqrt{1 - \left(-\frac{2}{3}\right)^2}\) and \(\sin(\theta) = -\sqrt{1 - \left(-\frac{2}{3}\right)^2}\).
Step 7 ：However, since we know that the sine of an angle in the second quadrant is positive, we discard the negative solution.
Step 8 ：Therefore, the sine of the angle is \(\boxed{0.745}\).
