Questions: First use the Pythagorean theorem to find the exact length of the missing side of the right triangle. Then find the exact values of the six trigonometric functions for the angle θ opposite the shortest side. Leg = 6√13 in, Hypotenuse = 6√17 in Part 1 of 2 Hypotenuse = 6√17 in, Leg = 6√13 in, Leg = 12 in.

Solution Steps

Using the Pythagorean theorem, we can find the length of the missing side \( b \) of the right triangle. Given the lengths of one leg \( a = 6\sqrt{13} \) and the hypotenuse \( c = 6\sqrt{17} \), we have:

\[ c^2 = a^2 + b^2 \]

Calculating \( b \):

\[ b = \sqrt{c^2 - a^2} = \sqrt{(6\sqrt{17})^2 - (6\sqrt{13})^2} = \sqrt{(36 \cdot 17) - (36 \cdot 13)} = \sqrt{612 - 468} = \sqrt{144} = 12 \]

Next, we calculate the six trigonometric functions for the angle \( \theta \) opposite the shortest side \( a \).

1. Sine: \[ \sin(\theta) = \frac{a}{c} = \frac{6\sqrt{13}}{6\sqrt{17}} = \frac{\sqrt{13}}{\sqrt{17}} = \frac{\sqrt{221}}{17} \]

2. Cosine: \[ \cos(\theta) = \frac{b}{c} = \frac{12}{6\sqrt{17}} = \frac{2}{\sqrt{17}} = \frac{2\sqrt{17}}{17} \]

3. Tangent: \[ \tan(\theta) = \frac{a}{b} = \frac{6\sqrt{13}}{12} = \frac{\sqrt{13}}{2} \]

4. Cosecant: \[ \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{1}{\frac{\sqrt{221}}{17}} = \frac{17}{\sqrt{221}} = \frac{\sqrt{221}}{13} \]

5. Secant: \[ \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{\frac{2\sqrt{17}}{17}} = \frac{17}{2\sqrt{17}} = \frac{\sqrt{17}}{2} \]

6. Cotangent: \[ \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{2}{\sqrt{13}} = \frac{2\sqrt{13}}{13} \]

The exact values of the six trigonometric functions for the angle \( \theta \) are:

• \( \sin(\theta) = \frac{\sqrt{221}}{17} \)
• \( \cos(\theta) = \frac{2\sqrt{17}}{17} \)
• \( \tan(\theta) = \frac{\sqrt{13}}{2} \)
• \( \csc(\theta) = \frac{\sqrt{221}}{13} \)
• \( \sec(\theta) = \frac{\sqrt{17}}{2} \)
• \( \cot(\theta) = \frac{2\sqrt{13}}{13} \)

Final Answer