Practice worksheet for calculating trigonometric ratios using right triangles.

Let’s go through each problem one by one.

We are given six right triangles, and for each, we’re asked to find a specific trigonometric ratio (sin, cos, tan) of a labeled angle. Since these are right triangles, we can use SOH-CAH-TOA:

- sin(θ) = opposite / hypotenuse
- cos(θ) = adjacent / hypotenuse
- tan(θ) = opposite / adjacent

We’ll identify the sides relative to the labeled angle in each triangle.

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Problem 1: Find sin X

Triangle: Right triangle with right angle at Y. Angle X is at bottom left. Side XY = 12, YZ = 5, XZ = 13 (hypotenuse).

Angle X:
- Opposite side = YZ = 5
- Hypotenuse = XZ = 13
→ sin X = 5/13

✔ Check: 5² + 12² = 25 + 144 = 169 = 13² → valid right triangle.

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Problem 2: Find cos G

Triangle: Right triangle with right angle at F. Angle G is at top. Side FG = 12, FH = 9, GH = 15 (hypotenuse).

Angle G:
- Adjacent side = FG = 12
- Hypotenuse = GH = 15
→ cos G = 12/15 = 4/5 (simplified)

✔ Check: 9² + 12² = 81 + 144 = 225 = 15² → valid.

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Problem 3: Find tan T

Triangle: Right triangle with right angle at R. Angle T is at bottom right. Side RT = 24, RS = 7, ST = 25 (hypotenuse).

Angle T:
- Opposite side = RS = 7
- Adjacent side = RT = 24
→ tan T = 7/24

✔ Check: 7² + 24² = 49 + 576 = 625 = 25² → valid.

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Problem 4: Find sin U

Triangle: Right triangle with right angle at S. Angle U is at bottom right. Side SU = 24, ST = 10, TU = 26 (hypotenuse).

Angle U:
- Opposite side = ST = 10
- Hypotenuse = TU = 26
→ sin U = 10/26 = 5/13 (simplified)

✔ Check: 10² + 24² = 100 + 576 = 676 = 26² → valid.

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Problem 5: Find cos V

Triangle: Right triangle with right angle at T. Angle V is at bottom right. Side TV = 12, TU = 5, UV = 13 (hypotenuse).

Angle V:
- Adjacent side = TV = 12
- Hypotenuse = UV = 13
→ cos V = 12/13

✔ Check: 5² + 12² = 25 + 144 = 169 = 13² → valid.

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Problem 6: Find tan P

Triangle: Right triangle with right angle at N. Angle P is at top. Side NP = 24, NM = 10, MP = 26 (hypotenuse).

Angle P:
- Opposite side = NM = 10
- Adjacent side = NP = 24
→ tan P = 10/24 = 5/12 (simplified)

✔ Check: 10² + 24² = 100 + 576 = 676 = 26² → valid.

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Final Answers:

1. sin X = 5/13
2. cos G = 4/5
3. tan T = 7/24
4. sin U = 5/13
5. cos V = 12/13
6. tan P = 5/12

Final Answer:
1. 5/13
2. 4/5
3. 7/24
4. 5/13
5. 12/13
6. 5/12