Trigonometry worksheet with 12 right triangle problems to find missing angles and side lengths.

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

Remember:
- sin(x) = opposite / hypotenuse
- cos(x) = adjacent / hypotenuse
- tan(x) = opposite / adjacent

We’ll also use inverse functions (like sin⁻¹, cos⁻¹, tan⁻¹) to find angles when we know the side ratios.

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Section A: Find the missing angle x

#### Problem 1:
Right triangle with:
- Opposite = 4 cm
- Adjacent = 9 cm
→ Use tan(x) = opposite/adjacent = 4/9

Calculate:
x = tan⁻¹(4 ÷ 9) = tan⁻¹(0.444...) ≈ 23.96° → Round to 3 sig figs: 24.0°

✔ Check: tan(24.0°) ≈ 0.445 → close enough.

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#### Problem 2:
Right triangle with:
- Hypotenuse = 14 cm
- Adjacent = 10 cm
→ Use cos(x) = adjacent/hypotenuse = 10/14

Calculate:
x = cos⁻¹(10 ÷ 14) = cos⁻¹(0.714...) ≈ 44.415° → Round to 3 sig figs: 44.4°

✔ Check: cos(44.4°) ≈ 0.714 → correct.

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#### Problem 3:
Right triangle with:
- Opposite = 7 cm
- Adjacent = 12 cm
→ Use tan(x) = 7/12

Calculate:
x = tan⁻¹(7 ÷ 12) = tan⁻¹(0.5833...) ≈ 30.256° → Round to 3 sig figs: 30.3°

✔ Check: tan(30.3°) ≈ 0.584 → good.

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#### Problem 4:
Right triangle with:
- Hypotenuse = 18 cm
- Adjacent = 15 cm
→ Use cos(x) = 15/18 = 5/6

Calculate:
x = cos⁻¹(5 ÷ 6) = cos⁻¹(0.8333...) ≈ 33.557° → Round to 3 sig figs: 33.6°

✔ Check: cos(33.6°) ≈ 0.833 → correct.

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#### Problem 5:
Right triangle with:
- Opposite = 0.98 cm
- Adjacent = 0.83 cm
→ Use tan(x) = 0.98 / 0.83

Calculate:
x = tan⁻¹(0.98 ÷ 0.83) = tan⁻¹(1.1807...) ≈ 49.74° → Round to 3 sig figs: 49.7°

✔ Check: tan(49.7°) ≈ 1.18 → good.

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#### Problem 6:
Right triangle with:
- Opposite = 410 mm
- Hypotenuse = 972 mm
→ Use sin(x) = 410 / 972

Calculate:
x = sin⁻¹(410 ÷ 972) = sin⁻¹(0.4218...) ≈ 24.95° → Round to 3 sig figs: 25.0°

✔ Check: sin(25.0°) ≈ 0.4226 → very close.

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Section B: Find missing angles or lengths x

#### Problem 1:
Right triangle with:
- Opposite = 5 cm
- Adjacent = 11 cm
→ Use tan(x) = 5/11

Calculate:
x = tan⁻¹(5 ÷ 11) = tan⁻¹(0.4545...) ≈ 24.44° → Round to 3 sig figs: 24.4°

✔ Check: tan(24.4°) ≈ 0.454 → correct.

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#### Problem 2:
Right triangle with:
- Hypotenuse = 0.7 cm
- Adjacent = 0.21 cm
→ Use cos(x) = 0.21 / 0.7 = 0.3

Calculate:
x = cos⁻¹(0.3) ≈ 72.54° → Round to 3 sig figs: 72.5°

✔ Check: cos(72.5°) ≈ 0.3007 → good.

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#### Problem 3:
Right triangle with:
- Angle = 62°
- Adjacent = 6.4 cm
- Need to find hypotenuse (x)
→ Use cos(62°) = adjacent / hypotenuse = 6.4 / x

So:
x = 6.4 / cos(62°)

cos(62°) ≈ 0.4695
x = 6.4 ÷ 0.4695 ≈ 13.63 cm → Round to 3 sig figs: 13.6 cm

✔ Check: 6.4 / 13.6 ≈ 0.4706 → cos(62°) ≈ 0.4695 → close enough.

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#### Problem 4:
Right triangle with:
- Hypotenuse = 27 mm
- Opposite = 1.5 cm → Wait! Units mismatch!

Convert 1.5 cm to mm: 1.5 cm = 15 mm

Now:
- Opposite = 15 mm
- Hypotenuse = 27 mm
→ Use sin(x) = 15 / 27 = 5/9

Calculate:
x = sin⁻¹(5 ÷ 9) = sin⁻¹(0.555...) ≈ 33.75° → Round to 3 sig figs: 33.8°

✔ Check: sin(33.8°) ≈ 0.556 → good.

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#### Problem 5:
Right triangle with:
- Angle = 15°
- Adjacent = 2.3 cm
- Need to find opposite (x)
→ Use tan(15°) = opposite / adjacent = x / 2.3

So:
x = 2.3 × tan(15°)

tan(15°) ≈ 0.2679
x = 2.3 × 0.2679 ≈ 0.616 cm → Round to 3 sig figs: 0.616 cm

✔ Check: 0.616 / 2.3 ≈ 0.2678 → matches tan(15°).

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#### Problem 6:
Right triangle with:
- Angle = ? (not given)
- Adjacent = 620 cm
- Hypotenuse = 8 m → Convert to same unit!

8 m = 800 cm

Now:
- Adjacent = 620 cm
- Hypotenuse = 800 cm
→ Use cos(x) = 620 / 800 = 0.775

Calculate:
x = cos⁻¹(0.775) ≈ 39.18° → Round to 3 sig figs: 39.2°

✔ Check: cos(39.2°) ≈ 0.775 → perfect.

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Final Answer:

Section A:
1) x = 24.0°
2) x = 44.4°
3) x = 30.3°
4) x = 33.6°
5) x = 49.7°
6) x = 25.0°

Section B:
1) x = 24.4°
2) x = 72.5°
3) x = 13.6 cm
4) x = 33.8°
5) x = 0.616 cm
6) x = 39.2°