Question $\newline$ Given $\tan A=-\frac{11}{60}$ and that angle $A$ is in Quadrant IV, find the exact value of $\csc A$ in simplest radical form using a rational denominator. $\newline$ Answer Attempt $1$ out of $2$

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Find trigonometric ratios using a Pythagorean or reciprocal identity

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Q. Question

\newline

Given

\tan A=-\frac{11}{60}

and that angle

A

is in Quadrant IV, find the exact value of

\csc A

in simplest radical form using a rational denominator.

\newline

Answer Attempt

1

out of

2

Recall Tangent Definition: Recall the definition of the tangent function in terms of sine and cosine: $\tan A = \frac{\sin A}{\cos A}$ . Since $\tan A$ is negative in Quadrant IV and cosine is positive in Quadrant IV, sine must be negative. We also know that $\sin^2 A + \cos^2 A = 1$ , which is the Pythagorean identity.
Representing sin and cos: Given $\tan A = -\frac{11}{60}$ , we can represent $\sin A$ and $\cos A$ as $\sin A = -\frac{11}{k}$ and $\cos A = \frac{60}{k}$ , where $k$ is the hypotenuse of the right triangle formed by the angle $A$ . We need to find the value of $k$ using the Pythagorean identity.
Finding Value of k: Using the Pythagorean identity $\sin^2 A + \cos^2 A = 1$ , we substitute the values of $\sin A$ and $\cos A$ to find $k$ : $\newline$ $(-\frac{11}{k})^2 + (\frac{60}{k})^2 = 1$ $\newline$ $\frac{121}{k^2} + \frac{3600}{k^2} = 1$ $\newline$ $\frac{3721}{k^2} = 1$ $\newline$ $k^2 = 3721$ $\newline$ $k = \sqrt{3721}$ $\newline$ $k = 61$ , since $k$ is the hypotenuse and must be positive.
Calculating csc A: Now that we have $k$ , we can find $\sin A = -\frac{11}{k} = -\frac{11}{61}$ . The cosecant function is the reciprocal of the sine function, so $\csc A = \frac{1}{\sin A}$ .
Calculating $csc A$ : Now that we have $k$ , we can find $\sin A = -\frac{11}{k} = -\frac{11}{61}$ . The cosecant function is the reciprocal of the sine function, so $csc A = \frac{1}{\sin A}$ .Calculate $csc A$ using the value of $\sin A$ : $csc A = \frac{1}{(-\frac{11}{61})}$ $csc A = -\frac{61}{11}$ Since we are looking for the exact value in simplest radical form with a rational denominator, we need to check if the sine value can be expressed in radical form. However, since $-\frac{11}{61}$ is already in simplest form and does not involve a radical, this is the final answer.