Trigonometic Ratios (SOH CAH TOA): Missing Side Lengths and Angle ... - Free Printable

Let’s solve each problem step by step. We’ll use trigonometry — specifically, sine, cosine, and tangent ratios — to find missing angles or sides in right triangles.

Remember:
- SOH CAH TOA
- Sin = Opposite / Hypotenuse
- Cos = Adjacent / Hypotenuse
- Tan = Opposite / Adjacent

We’ll also make sure all units are the same before calculating (like converting mm to cm if needed).

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Section A: Find the missing angle x

#### Problem 1)
Right triangle with:
- Opposite side to x = 4 cm
- Hypotenuse = 9 cm

Use sin(x) = opposite/hypotenuse = 4/9

→ x = sin⁻¹(4/9) ≈ sin⁻¹(0.4444) ≈ 26.4°

✔ Check: 4 ÷ 9 = 0.444… → inverse sin on calculator gives ~26.387° → round to 3 sig figs → 26.4°

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#### Problem 2)
Adjacent = 10 cm, Hypotenuse = 14 cm

Use cos(x) = adjacent/hypotenuse = 10/14 ≈ 0.7143

→ x = cos⁻¹(0.7143) ≈ 44.4°

✔ Check: 10 ÷ 14 = 0.714285… → inverse cos → ~44.415° → 44.4°

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#### Problem 3)
Opposite = 7 cm, Adjacent = 12 cm

Use tan(x) = opposite/adjacent = 7/12 ≈ 0.5833

→ x = tan⁻¹(0.5833) ≈ 30.3°

✔ Check: 7 ÷ 12 = 0.5833… → inverse tan → ~30.256° → 30.3°

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#### Problem 4)
This is a bit tricky — angle x is at the top left. The side labeled 18 cm is *opposite* to x, and 15 cm is *adjacent*.

So again: tan(x) = opposite/adjacent = 18/15 = 1.2

→ x = tan⁻¹(1.2) ≈ 50.2°

✔ Check: 18 ÷ 15 = 1.2 → inverse tan → ~50.194° → 50.2°

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#### Problem 5)
Angle x is at the top. Side next to it (adjacent) = 0.99 cm, opposite = 0.83 cm

So: tan(x) = opposite/adjacent = 0.83 / 0.99 ≈ 0.8384

→ x = tan⁻¹(0.8384) ≈ 40.0°

✔ Check: 0.83 ÷ 0.99 ≈ 0.83838 → inverse tan → ~39.97° → rounds to 40.0° (3 sig figs)

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#### Problem 6)
Units are in mm — that’s fine as long as we’re consistent.

Angle x is at the top right. Side opposite to x = 410 mm, hypotenuse = 972 mm

Use sin(x) = opposite/hypotenuse = 410 / 972 ≈ 0.4218

→ x = sin⁻¹(0.4218) ≈ 24.9°

✔ Check: 410 ÷ 972 ≈ 0.4218 → inverse sin → ~24.94° → 24.9°

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Section B: Find missing angles or lengths x

#### Problem 1)
Opposite = 5 cm, Adjacent = 11 cm → find angle x

tan(x) = 5/11 ≈ 0.4545

→ x = tan⁻¹(0.4545) ≈ 24.4°

✔ Check: 5 ÷ 11 ≈ 0.4545 → inverse tan → ~24.44° → 24.4°

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#### Problem 2)
Adjacent = 0.21 cm, Hypotenuse = 0.7 cm → find angle x

cos(x) = 0.21 / 0.7 = 0.3

→ x = cos⁻¹(0.3) ≈ 72.5°

✔ Check: 0.21 ÷ 0.7 = 0.3 → inverse cos → ~72.542° → 72.5°

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#### Problem 3)
Given angle = 62°, adjacent side = ? (x), opposite = 6.4 cm

We need to find adjacent side → use tan(62°) = opposite/adjacent = 6.4 / x

So:
x = 6.4 / tan(62°)

tan(62°) ≈ 1.8807

→ x ≈ 6.4 / 1.8807 ≈ 3.40 cm

✔ Check: 6.4 ÷ 1.8807 ≈ 3.403 → 3.40 cm (3 sig figs)

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#### Problem 4)
Units mixed! 1.5 cm and 27 mm → convert 27 mm to 2.7 cm

Angle x is at top left. Opposite = 2.7 cm, Adjacent = 1.5 cm

tan(x) = 2.7 / 1.5 = 1.8

→ x = tan⁻¹(1.8) ≈ 60.9°

✔ Check: 2.7 ÷ 1.5 = 1.8 → inverse tan → ~60.945° → 60.9°

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#### Problem 5)
Given angle = 13°, opposite = 2.3 cm, find hypotenuse? Wait — no, look:

Actually, angle is at top, so side opposite to 13° is 2.3 cm, and we want side adjacent to 13°? Wait — let's label:

Triangle has:
- Angle at top = 13°
- Bottom side (horizontal) = 2.3 cm → this is *adjacent* to the 13° angle
- Vertical side = x → this is *opposite* to the 13° angle

Wait — actually, looking at diagram: vertical side is labeled x, horizontal base is 2.3 cm, angle at top is 13°.

So from the top angle:
- Opposite side = 2.3 cm (base)
- Adjacent side = x (vertical leg)

So: tan(13°) = opposite/adjacent = 2.3 / x

→ x = 2.3 / tan(13°)

tan(13°) ≈ 0.2309

→ x ≈ 2.3 / 0.2309 ≈ 9.96 cm

✔ Check: 2.3 ÷ 0.2309 ≈ 9.961 → 9.96 cm

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#### Problem 6)
Units mixed: 8 m and 620 cm → convert 620 cm to 6.2 m

Angle x is at top right. Side opposite to x = 6.2 m, hypotenuse = 8 m

sin(x) = 6.2 / 8 = 0.775

→ x = sin⁻¹(0.775) ≈ 50.8°

✔ Check: 6.2 ÷ 8 = 0.775 → inverse sin → ~50.807° → 50.8°

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## Final Answer:

Section A:
1) x = 26.4°
2) x = 44.4°
3) x = 30.3°
4) x = 50.2°
5) x = 40.0°
6) x = 24.9°

Section B:
1) x = 24.4°
2) x = 72.5°
3) x = 3.40 cm
4) x = 60.9°
5) x = 9.96 cm
6) x = 50.8°