Trigonometry Worksheets | Free - CommonCoreSheets - Free Printable

You’ve uploaded a worksheet titled “Introduction to Sin, Cos, Tan”, which asks students to identify the Opposite, Adjacent, and Hypotenuse of Angle *a* in each of 15 right triangles.

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## 📌 Key Concepts:

In any right triangle, for a given acute angle (here labeled angle *a*):

- Opposite side: The side *across from* angle *a*.
- Adjacent side: The side *next to* angle *a*, but *not* the hypotenuse.
- Hypotenuse: The side *opposite the right angle* — always the longest side.

> ✔ Important: The labels “Opposite” and “Adjacent” depend on which angle you’re referencing — here, it’s angle *a*.

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Let’s go through each problem one by one and identify the three sides relative to angle *a*.

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🔢 Problem-by-Problem Solutions:

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1)
Triangle: Right triangle, angle *a* at bottom left.
Sides: vertical = 8, horizontal = 3, hypotenuse = √(8²+3²) = √73 (but not needed).
→ Opposite = 8 (across from angle *a*)
→ Adjacent = 3 (next to angle *a*, not hypotenuse)
→ Hypotenuse = √73 or just label as hypotenuse — but since it’s not given numerically, we can say “hypotenuse” or leave as is.
✔ Answer: Opposite=8, Adjacent=3, Hypotenuse=√73

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2)
Angle *a* at bottom left. Sides: adjacent=9, opposite=5, hypotenuse=10.3
→ Opposite = 5
→ Adjacent = 9
→ Hypotenuse = 10.3
✔ Answer: Opposite=5, Adjacent=9, Hypotenuse=10.3

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3)
Angle *a* at top right. Triangle is oriented with right angle at bottom left.
Side across from *a* = 5 → Opposite
Side next to *a* = 6 → Adjacent
Hypotenuse = 7.8
✔ Answer: Opposite=5, Adjacent=6, Hypotenuse=7.8

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4)
Angle *a* at bottom left.
Opposite = 8 (vertical)
Adjacent = 7 (horizontal)
Hypotenuse = 10.6
✔ Answer: Opposite=8, Adjacent=7, Hypotenuse=10.6

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5)
Angle *a* at bottom right.
Opposite = 6 (top side)
Adjacent = 6.7 (bottom side)
Hypotenuse = 3? Wait — that doesn’t make sense. Let’s check:
Actually, side labeled 3 is opposite angle *a*. Side labeled 6.7 is adjacent. Side labeled 6 is hypotenuse? But 3² + 6.7² ≠ 6² → likely typo or mislabeling.
But per diagram:
→ Opposite = 3 (across from *a*)
→ Adjacent = 6.7 (next to *a*)
→ Hypotenuse = 6 (longest side, even if mathematically inconsistent — go by diagram)
✔ Answer: Opposite=3, Adjacent=6.7, Hypotenuse=6

*(Note: In real math, 3² + 6.7² ≈ 9 + 44.89 = 53.89, sqrt≈7.34 ≠ 6 — so this triangle is not accurate, but for worksheet purposes, follow labels.)*

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6)
Angle *a* at bottom left.
Opposite = 5 (vertical)
Adjacent = 3 (horizontal)
Hypotenuse = 5.8
✔ Answer: Opposite=5, Adjacent=3, Hypotenuse=5.8

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7)
Angle *a* at top left.
Opposite = 7 (bottom side)
Adjacent = 8 (left side)
Hypotenuse = 10.6
✔ Answer: Opposite=7, Adjacent=8, Hypotenuse=10.6

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8)
Angle *a* at bottom right.
Opposite = 7 (vertical)
Adjacent = 3 (horizontal)
Hypotenuse = 7.6
✔ Answer: Opposite=7, Adjacent=3, Hypotenuse=7.6

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9)
Angle *a* at bottom right.
Opposite = 5 (vertical)
Adjacent = 7 (horizontal)
Hypotenuse = 8.6
✔ Answer: Opposite=5, Adjacent=7, Hypotenuse=8.6

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10)
Angle *a* at top left.
Opposite = 5 (bottom side)
Adjacent = 8 (right side)
Hypotenuse = 9.4
✔ Answer: Opposite=5, Adjacent=8, Hypotenuse=9.4

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11)
Angle *a* at bottom right.
Opposite = 5 (vertical)
Adjacent = 6 (horizontal)
Hypotenuse = 7.8
✔ Answer: Opposite=5, Adjacent=6, Hypotenuse=7.8

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12)
Angle *a* at top left.
Opposite = 5 (bottom side)
Adjacent = 6 (right side)
Hypotenuse = 8.5
✔ Answer: Opposite=5, Adjacent=6, Hypotenuse=8.5

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13)
Angle *a* at bottom right.
Opposite = 8 (vertical)
Adjacent = 6 (horizontal)
Hypotenuse = 10.0
✔ Answer: Opposite=8, Adjacent=6, Hypotenuse=10.0

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14)
Angle *a* at bottom left.
Opposite = 7 (vertical)
Adjacent = 3 (horizontal)
Hypotenuse = 7.6
✔ Answer: Opposite=7, Adjacent=3, Hypotenuse=7.6

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15)
Angle *a* at top left.
Opposite = 4 (bottom side)
Adjacent = 8 (vertical side)
Hypotenuse = 8.9
✔ Answer: Opposite=4, Adjacent=8, Hypotenuse=8.9

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## ✍️ Final Answer Sheet (Fill-in-the-blank):

Here’s how to fill the “Answers” column:

```
1. Opposite=8, Adjacent=3, Hypotenuse=√73
2. Opposite=5, Adjacent=9, Hypotenuse=10.3
3. Opposite=5, Adjacent=6, Hypotenuse=7.8
4. Opposite=8, Adjacent=7, Hypotenuse=10.6
5. Opposite=3, Adjacent=6.7, Hypotenuse=6
6. Opposite=5, Adjacent=3, Hypotenuse=5.8
7. Opposite=7, Adjacent=8, Hypotenuse=10.6
8. Opposite=7, Adjacent=3, Hypotenuse=7.6
9. Opposite=5, Adjacent=7, Hypotenuse=8.6
10. Opposite=5, Adjacent=8, Hypotenuse=9.4
11. Opposite=5, Adjacent=6, Hypotenuse=7.8
12. Opposite=5, Adjacent=6, Hypotenuse=8.5
13. Opposite=8, Adjacent=6, Hypotenuse=10.0
14. Opposite=7, Adjacent=3, Hypotenuse=7.6
15. Opposite=4, Adjacent=8, Hypotenuse=8.9
```

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## 💡 Teaching Tip:

This worksheet is designed to build foundational understanding before introducing sine, cosine, and tangent ratios:

- sin(a) = Opposite / Hypotenuse
- cos(a) = Adjacent / Hypotenuse
- tan(a) = Opposite / Adjacent

Once students master identifying these sides, they can easily compute trig ratios.

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✔ All problems solved! Let me know if you’d like the next step — computing sin/cos/tan for each angle.