Given cosecant, A, equals, minus, start fraction, 14, divided by, 9, end fractioncscA=− 
9
14
​	
  and that angle AA is in Quadrant IV, find the exact value of cotangent, AcotA in simplest radical form using a rational denominator.

To find the exact value of \( \cot A \), we will follow a systematic approach. Given that:

\[
\csc A = -\frac{14}{9}
\]

and that angle \( A \) is in Quadrant IV.

### Step 1: Use the identity of cosecant
We know the identity:
\[
\csc A = \frac{1}{\sin A}
\]
Thus,
\[
\sin A = -\frac{9}{14}
\]
We use the negative value because sine is negative in Quadrant IV.

### Step 2: Use the Pythagorean identity to find cosine
The Pythagorean identity is:
\[
\sin^2 A + \cos^2 A = 1
\]
Substitute \( \sin A = -\frac{9}{14} \) into this identity:
\[
\left( -\frac{9}{14} \right)^2 + \cos^2 A = 1
\]
\[
\frac{81}{196} + \cos^2 A = 1
\]
\[
\cos^2 A = 1 - \frac{81}{196}
\]
\[
\cos^2 A = \frac{196}{196} - \frac{81}{196}
\]
\[
\cos^2 A = \frac{115}{196}
\]
Take the square root of both sides:
\[
\cos A = \pm \frac{\sqrt{115}}{14}
\]
Since cosine is positive in Quadrant IV, we take the positive value:
\[
\cos A = \frac{\sqrt{115}}{14}
\]

### Step 3: Use the definition of cotangent
The cotangent is defined as:
\[
\cot A = \frac{\cos A}{\sin A}
\]
Substitute \( \cos A = \frac{\sqrt{115}}{14} \) and \( \sin A = -\frac{9}{14} \):
\[
\cot A = \frac{\frac{\sqrt{115}}{14}}{-\frac{9}{14}} = \frac{\sqrt{115}}{-9}
\]
Thus, the exact value of \( \cot A \) is:
\[
\boxed{-\frac{\sqrt{115}}{9}}
\]

Would you like more details on any of the steps?

### Related Questions:
1. What is the exact value of \( \sec A \) given \( \csc A = -\frac{14}{9} \)?
2. How would the value of \( \cot A \) change if \( A \) were in Quadrant II?
3. Can you derive the value of \( \tan A \) from the given information?
4. How does the identity \( \cot A = \frac{1}{\tan A} \) apply here?
5. What other trigonometric identities can be derived from the given cosecant value?

### Tip:
Always keep track of the signs of trigonometric functions based on the quadrant the angle is located in.

Given cosecant A = -14/9 and that angle A is in Quadrant IV, find the exact value of cotangent A in simplest radical form using a rational denominator.

Solve a trigonometric problem involving cosecant A = -14/9 in Quadrant IV. Learn how to use trigonometric identities, such as the Pythagorean identity, to find the exact value of cotangent A in simplest radical form with a rational denominator. Suitable for high school students in Grades 11-12.

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