Problem

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 \(\boxed{0.745}\).

From Solvely APP
Source: https://solvelyapp.com/problems/7920/

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