for each of the following, calculate the unknown side using trig ratios
triangle A= hyp=10cm angle=30 degrees
triangle B= hyp=7c, angle=45 degres

To find the unknown sides of each triangle using trigonometric ratios, we'll apply the sine and cosine functions based on the given angles and the hypotenuse.

### Triangle A
- **Hypotenuse** (\(h\)) = 10 cm
- **Angle** (\(\theta\)) = 30 degrees

We can calculate both the adjacent side (\(a\)) and the opposite side (\(o\)) using the trigonometric ratios:

1. **Opposite Side (\(o\))**:
   \[
   \sin(\theta) = \frac{o}{h} \quad \Rightarrow \quad o = h \times \sin(\theta)
   \]
   \[
   o = 10 \times \sin(30^\circ) = 10 \times 0.5 = 5 \, \text{cm}
   \]

2. **Adjacent Side (\(a\))**:
   \[
   \cos(\theta) = \frac{a}{h} \quad \Rightarrow \quad a = h \times \cos(\theta)
   \]
   \[
   a = 10 \times \cos(30^\circ) = 10 \times \frac{\sqrt{3}}{2} \approx 10 \times 0.866 = 8.66 \, \text{cm}
   \]

### Triangle B
- **Hypotenuse** (\(h\)) = 7 cm
- **Angle** (\(\theta\)) = 45 degrees

Again, we'll calculate the adjacent side (\(a\)) and the opposite side (\(o\)):

1. **Opposite Side (\(o\))**:
   \[
   \sin(\theta) = \frac{o}{h} \quad \Rightarrow \quad o = h \times \sin(\theta)
   \]
   \[
   o = 7 \times \sin(45^\circ) = 7 \times \frac{\sqrt{2}}{2} \approx 7 \times 0.707 = 4.95 \, \text{cm}
   \]

2. **Adjacent Side (\(a\))**:
   \[
   \cos(\theta) = \frac{a}{h} \quad \Rightarrow \quad a = h \times \cos(\theta)
   \]
   \[
   a = 7 \times \cos(45^\circ) = 7 \times \frac{\sqrt{2}}{2} \approx 7 \times 0.707 = 4.95 \, \text{cm}
   \]

### Summary:
- **Triangle A**: Opposite side = 5 cm, Adjacent side = 8.66 cm
- **Triangle B**: Opposite side = 4.95 cm, Adjacent side = 4.95 cm

Would you like more details on any of these calculations?

### Related Questions:
1. How do you calculate the hypotenuse if the sides are known?
2. What is the sine and cosine of 60 degrees?
3. How does the tangent ratio relate to sine and cosine?
4. How would the side lengths change if the angle was 60 degrees instead of 30 degrees?
5. Can you solve for angles using the inverse trigonometric functions?

**Tip:** Remember that for a 45-degree angle, both the adjacent and opposite sides are equal in a right triangle.

for each of the following, calculate the unknown side using trig ratios: triangle A= hyp=10cm angle=30 degrees; triangle B= hyp=7cm, angle=45 degrees

Learn how to calculate the unknown sides of right triangles using trigonometric ratios. This example covers a triangle with a 30-degree angle and a hypotenuse of 10 cm, as well as a triangle with a 45-degree angle and a hypotenuse of 7 cm. Suitable for students in Grades 8-10.

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