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The image shows a right triangle with a hypotenuse of 15, one leg of length 5, and an unknown base \( a \). The goal is to solve for:

1. \( a \)
2. \( b = \tan(\theta) \)
3. \( c = \cos(\theta) \)
4. \( d = \sin(\theta) \)
5. \( e = \tan^{-1}(\theta) \) in degrees
6. \( f = \tan^{-1}(\theta) \) in radians

### Step 1: Finding \( a \) using the Pythagorean theorem
\[
a^2 + 5^2 = 15^2
\]
\[
a^2 + 25 = 225
\]
\[
a^2 = 225 - 25 = 200
\]
\[
a = \sqrt{200} = 10\sqrt{2} \approx 14.14
\]

### Step 2: Finding \( \theta \) using trigonometric ratios

- \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{5}{15} = \frac{1}{3} \)
- \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{a}{15} = \frac{10\sqrt{2}}{15} \)
\[
\cos(\theta) \approx \frac{14.14}{15} = 0.9428
\]

- \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{5}{a} = \frac{5}{10\sqrt{2}} \approx 0.3536 \)

### Step 3: Finding \( \theta \) in degrees and radians

Using the arctangent:
\[
\theta = \tan^{-1}(0.3536)
\]

Let me calculate the value of \( \theta \) in degrees and radians.It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can still calculate the inverse tangent manually or by using a calculator for the last two parts.

Would you like further details on any of the steps so far? Here are some follow-up questions for you:

1. How would the value of \( a \) change if the hypotenuse length was different?
2. Can you explain the relationship between the sine and cosine values in a right triangle?
3. What is the significance of converting \( \theta \) into radians?
4. How would you calculate the cotangent of \( \theta \)?
5. What tools can you use to verify trigonometric identities?

**Tip:** When using inverse trigonometric functions, make sure to use the appropriate mode on your calculator (degrees or radians) for accurate results.

Find the unknown side of a right triangle with a hypotenuse of 15 and one leg of length 5. Also, calculate trigonometric functions for the angle and its inverse in degrees and radians.

Learn how to solve a right triangle with a hypotenuse of 15 and one leg of 5. Use the Pythagorean theorem and trigonometric functions (sine, cosine, tangent) to calculate side lengths and angles. Includes inverse tangent calculations in degrees and radians.

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