Given that ` cos ""(A)/(2) = (12)/(13),` calculate without the use of tubles, the values of sin A , cos A and tan A.

Answer

To solve the problem step by step, we will start with the given information and use trigonometric identities to find the values of sin A, cos A, and tan A.

### Step 1: Given Information
We are given that:
\[ \cos\left(\frac{A}{2}\right) = \frac{12}{13} \]

### Step 2: Use the Cosine Half-Angle Identity
We can use the cosine half-angle identity:
\[ \cos A = 2\cos^2\left(\frac{A}{2}\right) - 1 \]
Substituting the given value:
\[ \cos A = 2\left(\frac{12}{13}\right)^2 - 1 \]

### Step 3: Calculate \(\cos A\)
Calculating \(\left(\frac{12}{13}\right)^2\):
\[ \left(\frac{12}{13}\right)^2 = \frac{144}{169} \]
Now substituting back into the equation for \(\cos A\):
\[ \cos A = 2 \cdot \frac{144}{169} - 1 = \frac{288}{169} - 1 = \frac{288}{169} - \frac{169}{169} = \frac{119}{169} \]

### Step 4: Calculate \(\sin A\)
Using the Pythagorean identity:
\[ \sin^2 A + \cos^2 A = 1 \]
We can find \(\sin A\):
\[ \sin^2 A = 1 - \cos^2 A = 1 - \left(\frac{119}{169}\right)^2 \]
Calculating \(\left(\frac{119}{169}\right)^2\):
\[ \left(\frac{119}{169}\right)^2 = \frac{14161}{28561} \]
Now substituting back:
\[ \sin^2 A = 1 - \frac{14161}{28561} = \frac{28561 - 14161}{28561} = \frac{14300}{28561} \]
Taking the square root:
\[ \sin A = \sqrt{\frac{14300}{28561}} = \frac{\sqrt{14300}}{169} \]
Calculating \(\sqrt{14300}\):
\[ \sqrt{14300} = 120 \]
Thus:
\[ \sin A = \frac{120}{169} \]

### Step 5: Calculate \(\tan A\)
Using the definition of tangent:
\[ \tan A = \frac{\sin A}{\cos A} \]
Substituting the values we found:
\[ \tan A = \frac{\frac{120}{169}}{\frac{119}{169}} = \frac{120}{119} \]

### Final Values
Thus, we have:
\[ \sin A = \frac{120}{169}, \quad \cos A = \frac{119}{169}, \quad \tan A = \frac{120}{119} \]