Practice finding trigonometric ratios using segments in right triangles.

Explanation:
We are asked to find each trigonometric ratio using the given right triangles. For each triangle, we need to identify:

- Opposite side (relative to angle θ)
- Adjacent side (next to angle θ, not the hypotenuse)
- Hypotenuse (longest side, opposite the right angle)

Then use:
- sin θ = opposite / hypotenuse
- cos θ = adjacent / hypotenuse
- tan θ = opposite / adjacent

Let’s go one by one.

1) sin θ
Triangle has:
- Right angle at C
- Angle θ at A
- So:
- Opposite to θ = BC = 3
- Adjacent = AC = 4
- Hypotenuse = AB = 5 (since 3-4-5 triangle)
→ sin θ = 3/5

2) cos θ
Triangle: right angle at Z
Angle θ at X
So:
- Opposite = YZ = 4
- Adjacent = XZ = 3
- Hypotenuse = XY = 5
→ cos θ = adjacent / hypotenuse = 3/5

3) tan θ
Right angle at Z
Angle θ at Y
So:
- Opposite to θ = XZ = 5
- Adjacent = YZ = 12
- Hypotenuse = XY = 13 (5-12-13 triangle)
→ tan θ = opposite / adjacent = 5/12

4) sin θ
Right angle at R
Angle θ at Q
So:
- Opposite = PR = 8
- Adjacent = QR = 6
- Hypotenuse = PQ = 10 (6-8-10 = scaled 3-4-5)
→ sin θ = 8/10 = 4/5

5) cos θ
Right angle at S
Angle θ at R
So:
- Opposite = ST = 7
- Adjacent = RS = 24
- Hypotenuse = RT = 25 (7-24-25 triangle)
→ cos θ = adjacent / hypotenuse = 24/25

6) tan θ
Right angle at Z
Angle θ at X
So:
- Opposite = YZ = 9
- Adjacent = XZ = 12
- Hypotenuse = XY = 15 (9-12-15 = scaled 3-4-5)
→ tan θ = 9/12 = 3/4

Let me double-check each:

1) 3-4-5 → sin = 3/5 ✔
2) Same triangle, cos at X → adjacent is XZ = 3, hyp = 5 → 3/5 ✔
3) 5-12-13, angle at Y → opposite is XZ = 5, adjacent = YZ = 12 → 5/12 ✔
4) 6-8-10, angle at Q → opposite is PR = 8, hyp = 10 → 8/10 = 4/5 ✔
5) 7-24-25, angle at R → adjacent = RS = 24, hyp = 25 → 24/25 ✔
6) 9-12-15, angle at X → opposite = YZ = 9, adjacent = XZ = 12 → 9/12 = 3/4 ✔

All correct.

Final Answer:
1) 3/5
2) 3/5
3) 5/12
4) 4/5
5) 24/25
6) 3/4