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Let's solve the problem using the right triangle properties.

We are given:
- \( a = 7 \, \text{ft} \) (opposite side to angle \( \beta \))
- \( b = 12 \, \text{ft} \) (adjacent side to angle \( \beta \))

To find the sine, cosine, and tangent of \( \beta \), we need to calculate the hypotenuse \( c \) using the Pythagorean theorem:

\[
c = \sqrt{a^2 + b^2}
\]

\[
c = \sqrt{7^2 + 12^2} = \sqrt{49 + 144} = \sqrt{193}
\]

Now we can calculate the trigonometric functions:

1. **Sine of \( \beta \)**:
\[
\sin \beta = \frac{a}{c} = \frac{7}{\sqrt{193}}
\]

2. **Cosine of \( \beta \)**:
\[
\cos \beta = \frac{b}{c} = \frac{12}{\sqrt{193}}
\]

3. **Tangent of \( \beta \)**:
\[
\tan \beta = \frac{a}{b} = \frac{7}{12}
\]

### Exact values:
- \( \sin \beta = \frac{7}{\sqrt{193}} \)
- \( \cos \beta = \frac{12}{\sqrt{193}} \)
- \( \tan \beta = \frac{7}{12} \)

### Approximate values:
Let me calculate the approximations to three decimal places.### Approximate values (to three decimal places):
- \( \sin \beta \approx 0.504 \)
- \( \cos \beta \approx 0.864 \)
- \( \tan \beta \approx 0.583 \)

The hypotenuse \( c \approx 13.892 \, \text{ft} \).

Do you have any questions or would you like further details on this?

Here are 5 related questions to deepen your understanding:
1. How does the Pythagorean theorem apply to right triangles?
2. What are the definitions of sine, cosine, and tangent for a right triangle?
3. How can you convert trigonometric values to angles using inverse trigonometric functions?
4. What are the relationships between the angles in a right triangle?
5. How can we calculate trigonometric values if we only know one angle?

**Tip**: Always ensure that the triangle you're working with is a right triangle when applying trigonometric ratios!

Find sine, cosine, and tangent of β, given that a = 7 ft and b = 12 ft. Present both the exact value and an approximate value (accurate up to three or more decimals).

Learn how to calculate the sine, cosine, and tangent of angle β in a right triangle where a = 7 ft and b = 12 ft. This problem covers key trigonometric concepts and the Pythagorean theorem. Suitable for students in Grades 9-12.

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