https://file.aiquickdraw.com/mathbot/file/17260241852787wssda9r.jpg

Let's break down the two problems provided in the image:

### Problem 5:
We are given a right triangle where:
- The opposite side to angle \( \alpha \) is 3.
- The adjacent side to angle \( \alpha \) is 4.
- The hypotenuse is 5.

We are asked to evaluate \( \sin(\alpha) \), \( \cos(\alpha) \), and \( \tan(\alpha) \).

1. **Sine**:
   \[
   \sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5}
   \]
   
2. **Cosine**:
   \[
   \cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5}
   \]
   
3. **Tangent**:
   \[
   \tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{4}
   \]

---

### Problem 6:
We have a right triangle with:
- One known side \( b = 7 \) (adjacent side to the 30° angle).
- The angle \( 30^\circ \).

We need to find the unknown sides \( a \) (opposite) and \( c \) (hypotenuse).

We will use trigonometric ratios for a 30° angle:
1. **Sine**:
   \[
   \sin(30^\circ) = \frac{a}{c} = \frac{1}{2}
   \]
   Therefore, \( a = \frac{1}{2} \cdot c \).

2. **Cosine**:
   \[
   \cos(30^\circ) = \frac{7}{c} = \frac{\sqrt{3}}{2}
   \]
   Solving for \( c \):
   \[
   c = \frac{7 \times 2}{\sqrt{3}} = \frac{14}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{14\sqrt{3}}{3}
   \]

3. **Finding \( a \)**:
   Using the relationship \( a = \frac{1}{2} \cdot c \):
   \[
   a = \frac{1}{2} \cdot \frac{14\sqrt{3}}{3} = \frac{7\sqrt{3}}{3}
   \]

---

Let me know if you'd like further clarification or step-by-step details.

### Follow-up questions:
1. How would you calculate the sine, cosine, and tangent of a different angle in the triangle?
2. Can you use the Pythagorean theorem to verify the sides of the triangles in either problem?
3. How do trigonometric ratios change when the angle is larger or smaller than 30°?
4. What are the values of sine, cosine, and tangent for other common angles like 45° or 60°?
5. How would the calculations change if the triangle were not a right triangle?

**Tip**: Always remember that the sine and cosine of complementary angles add up to 1 (i.e., \( \sin(\theta) + \cos(90^\circ - \theta) = 1 \)).

Using the triangle shown in the picture, evaluate sin α, cos α, and tan α. Find the unknown sides of the triangle in another problem involving a 30° angle.

Learn how to calculate sine, cosine, and tangent for angle α in a right triangle. Also, find the unknown sides of a triangle with a 30° angle using trigonometric ratios. This problem is ideal for students in Grades 8-10.

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