https://file.aiquickdraw.com/mathbot/file/172840101642476ly54i7.jpg

Let's analyze both triangles from the image and solve for the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) for the given angles.

### 1. **First Triangle: Angle \( \alpha \)**
For the first triangle, we are given:
- Adjacent side to \( \alpha \): \( 31.68 \)
- Hypotenuse: \( 42.71 \)

We can use the Pythagorean theorem to calculate the opposite side \( \text{opposite} = \sqrt{42.71^2 - 31.68^2} \).

#### Step 1: Calculate opposite side
\[
\text{opposite} = \sqrt{42.71^2 - 31.68^2} = \sqrt{1824.9441 - 1003.1424} = \sqrt{821.8017} \approx 28.66
\]

#### Step 2: Calculate the six trigonometric functions for \( \alpha \)

- **Sine** \( \sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{28.66}{42.71} \approx 0.671 \)
- **Cosine** \( \cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{31.68}{42.71} \approx 0.742 \)
- **Tangent** \( \tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}} = \frac{28.66}{31.68} \approx 0.905 \)
- **Cosecant** \( \csc(\alpha) = \frac{1}{\sin(\alpha)} = \frac{1}{0.671} \approx 1.490 \)
- **Secant** \( \sec(\alpha) = \frac{1}{\cos(\alpha)} = \frac{1}{0.742} \approx 1.348 \)
- **Cotangent** \( \cot(\alpha) = \frac{1}{\tan(\alpha)} = \frac{1}{0.905} \approx 1.105 \)

### 2. **Second Triangle: Angle \( B \)**
For the second triangle, we are given:
- Opposite side to \( B \): \( 1 \)
- Adjacent side to \( B \): \( \sqrt{2} \)
- Hypotenuse \( c \): This can be calculated using the Pythagorean theorem.

#### Step 1: Calculate the hypotenuse
\[
c = \sqrt{(\sqrt{2})^2 + 1^2} = \sqrt{2 + 1} = \sqrt{3} \approx 1.732
\]

#### Step 2: Calculate the six trigonometric functions for \( B \)

- **Sine** \( \sin(B) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{1.732} \approx 0.577 \)
- **Cosine** \( \cos(B) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\sqrt{2}}{1.732} \approx 0.816 \)
- **Tangent** \( \tan(B) = \frac{\text{opposite}}{\text{adjacent}} = \frac{1}{\sqrt{2}} \approx 0.707 \)
- **Cosecant** \( \csc(B) = \frac{1}{\sin(B)} = \frac{1}{0.577} \approx 1.732 \)
- **Secant** \( \sec(B) = \frac{1}{\cos(B)} = \frac{1}{0.816} \approx 1.225 \)
- **Cotangent** \( \cot(B) = \frac{1}{\tan(B)} = \frac{1}{0.707} \approx 1.414 \)

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Would you like more details on any part of the process or a breakdown of these functions? Here are five questions to extend this topic:

1. How do the trigonometric functions change as the angle approaches 90°?
2. What are the real-world applications of these trigonometric functions?
3. How can the Pythagorean theorem be applied in non-right triangles?
4. What is the relationship between the trigonometric functions of complementary angles?
5. How can you calculate angles if you are only given the sides of a triangle?

**Tip:** Always remember that sine and cosine values are between -1 and 1, helping you check if your calculations are reasonable!

Calculate the six trigonometric functions for angle α in the first triangle and for angle B in the second triangle.

Solve for the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) of angle α and angle B in two given right triangles using trigonometric ratios and the Pythagorean Theorem. Suitable for students in Grades 9-12.

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