Right Triangle AIS has hypotenuse 117 side a of 6 and side b is unknown. find the exact value of tangent of angle AIS.

Given the right triangle \( \triangle AIS \) with:
- Hypotenuse (\( c \)) = 117
- Side \( a \) = 6
- Side \( b \) is unknown.

We will find the exact value of \( \tan(\angle AIS) \).

### Step 1: Find Side \( b \)

Using the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
Substituting the given values:
\[
6^2 + b^2 = 117^2
\]
\[
36 + b^2 = 13689
\]
\[
b^2 = 13689 - 36 = 13653
\]
\[
b = \sqrt{13653}
\]

### Step 2: Simplify \( \sqrt{13653} \)

The exact value of \( \sqrt{13653} \) cannot be simplified easily as a perfect square, so we'll keep it in its square root form:
\[
b = \sqrt{13653}
\]

### Step 3: Calculate \( \tan(\angle AIS) \)

The tangent of angle \( AIS \) is given by:
\[
\tan(\angle AIS) = \frac{\text{opposite}}{\text{adjacent}} = \frac{b}{a}
\]
\[
\tan(\angle AIS) = \frac{\sqrt{13653}}{6}
\]

### Exact Value
The exact value of \( \tan(\angle AIS) \) is:
\[
\tan(\angle AIS) = \frac{\sqrt{13653}}{6}
\]

Would you like more details or have any questions about these calculations?

Here are 5 related questions that might interest you:
1. How do you find the sine and cosine of angle \( AIS \) in this triangle?
2. What is the approximate decimal value of \( \tan(\angle AIS) \)?
3. How can we simplify the expression \( \sqrt{13653} \) further?
4. How do we determine the values of all angles in a right triangle using trigonometric ratios?
5. What are some real-life applications of using trigonometric ratios in right triangles?

**Tip:** Knowing the trigonometric ratios for special angles can speed up calculations for certain problems.

Right Triangle AIS has hypotenuse 117, side a of 6, and side b is unknown. Find the exact value of the tangent of angle AIS.

Explore how to find the exact value of the tangent of angle AIS in a right triangle with hypotenuse 117 and side a = 6. This problem covers trigonometric functions and the Pythagorean theorem, offering a step-by-step solution. Suitable for students in Grades 9-11.

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