If cosA`=(4)/(5) and angleA` is not is Quadrant I, what is the value of `sinA`?

Options

A. `-0.6`
B. `-0.2`
C. `0.6`
D. `0.75`

Answer

To find the value of \( \sin A \) given that \( \cos A = \frac{4}{5} \) and angle \( A \) is not in Quadrant I, we can follow these steps:

### Step 1: Use the Pythagorean Identity
We know from the Pythagorean identity that:
\[ \sin^2 A + \cos^2 A = 1 \]
This means we can express \( \sin^2 A \) in terms of \( \cos^2 A \):
\[ \sin^2 A = 1 - \cos^2 A \]

### Step 2: Calculate \( \cos^2 A \)
Given \( \cos A = \frac{4}{5} \), we can calculate \( \cos^2 A \):
\[ \cos^2 A = \left(\frac{4}{5}\right)^2 = \frac{16}{25} \]

### Step 3: Substitute \( \cos^2 A \) into the Identity
Now we substitute \( \cos^2 A \) into the Pythagorean identity:
\[ \sin^2 A = 1 - \frac{16}{25} \]

### Step 4: Simplify the Expression
We need to simplify the right-hand side:
\[ \sin^2 A = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \]

### Step 5: Take the Square Root
Now we take the square root of both sides to find \( \sin A \):
\[ \sin A = \pm \sqrt{\frac{9}{25}} = \pm \frac{3}{5} \]

### Step 6: Determine the Sign of \( \sin A \)
Since we know that angle \( A \) is not in Quadrant I, it must be in either Quadrant II or Quadrant III. In these quadrants, the sine function is negative in Quadrant III and positive in Quadrant II. Since we are given that \( A \) is not in Quadrant I, we need to determine which quadrant it is in.

If \( A \) is in Quadrant II, \( \sin A \) would be positive, and if \( A \) is in Quadrant III, \( \sin A \) would be negative. Since the problem does not specify, but implies that it is not in Quadrant I, we will assume it is in Quadrant III for this case.

Thus, we take the negative value:
\[ \sin A = -\frac{3}{5} \]

### Final Answer
\[ \sin A = -\frac{3}{5} \]

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