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We are given that \(\sin(A) = 0.47\) and \(\cos(A) < 0\). Our goal is to calculate:

1. \(\cos(A)\)
2. \(\sin(2A)\)
3. \(\cos(2A)\)
4. \(\tan(2A)\)
5. The quadrant for \(2A\).

### Step 1: Find \(\cos(A)\)

Using the Pythagorean identity:

\[
\sin^2(A) + \cos^2(A) = 1
\]

Substitute \(\sin(A) = 0.47\):

\[
0.47^2 + \cos^2(A) = 1
\]

\[
0.2209 + \cos^2(A) = 1
\]

\[
\cos^2(A) = 1 - 0.2209 = 0.7791
\]

\[
\cos(A) = \pm \sqrt{0.7791} \approx \pm 0.8827
\]

Since \(\cos(A) < 0\) (from the problem), we select the negative value:

\[
\cos(A) = -0.8827
\]

### Step 2: Find \(\sin(2A)\)

Using the double angle formula for sine:

\[
\sin(2A) = 2 \sin(A) \cos(A)
\]

Substitute \(\sin(A) = 0.47\) and \(\cos(A) = -0.8827\):

\[
\sin(2A) = 2 \times 0.47 \times (-0.8827) = -0.8301
\]

### Step 3: Find \(\cos(2A)\)

Using the double angle formula for cosine:

\[
\cos(2A) = \cos^2(A) - \sin^2(A)
\]

Substitute \(\cos(A) = -0.8827\) and \(\sin(A) = 0.47\):

\[
\cos(2A) = (-0.8827)^2 - (0.47)^2 = 0.7791 - 0.2209 = 0.5582
\]

### Step 4: Find \(\tan(2A)\)

Using the double angle formula for tangent:

\[
\tan(2A) = \frac{\sin(2A)}{\cos(2A)}
\]

Substitute \(\sin(2A) = -0.8301\) and \(\cos(2A) = 0.5582\):

\[
\tan(2A) = \frac{-0.8301}{0.5582} \approx -1.4871
\]

### Step 5: Determine the Quadrant for \(2A\)

Since \(\sin(2A) = -0.8301\) and \(\cos(2A) = 0.5582\), \(\sin(2A)\) is negative and \(\cos(2A)\) is positive. This places the angle \(2A\) in the **fourth quadrant**.

### Final Answers:
1. \(\cos(A) = -0.8827\)
2. \(\sin(2A) = -0.8301\)
3. \(\cos(2A) = 0.5582\)
4. \(\tan(2A) = -1.4871\)
5. The quadrant for \(2A\) is the **fourth quadrant**.

Would you like further details or have any questions?

Here are 5 relative questions:
1. How do we derive the double angle formulas for sine and cosine?
2. What happens to the angle’s quadrant if both sine and cosine are positive?
3. How can we use trigonometric identities to solve for unknown angles?
4. What other methods can be used to solve for trigonometric values?
5. How do we identify the quadrant for \(A\) and \(2A\) using the unit circle?

**Tip:** Always remember to check the signs of trigonometric values when determining the correct quadrant.

If sin(A) = 0.47 and cos(A) < 0, determine: cos(A), sin(2A), cos(2A), tan(2A), and the quadrant for 2A.

Calculate the values of cos(A), sin(2A), cos(2A), and tan(2A) when sin(A) = 0.47 and cos(A) < 0. This problem covers trigonometric identities, including the Pythagorean identity and double angle formulas. Suitable for high school students in Grades 10-12.

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