Here is the complete solution to the
Trigonometry (B) - Missing Angles and Lengths worksheet, with explanations for each problem. All answers are given to
3 significant figures as requested.
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## 🔷 SECTION A: Find the missing angle x
All problems in Section A involve right-angled triangles. We use
trigonometric ratios:
-
SOH CAH TOA:
- sin(x) = Opposite / Hypotenuse
- cos(x) = Adjacent / Hypotenuse
- tan(x) = Opposite / Adjacent
We find the angle using the
inverse trig functions (sin⁻¹, cos⁻¹, tan⁻¹).
---
1)
- Opposite = 4 cm, Hypotenuse = 9 cm → Use
sin
- sin(x) = 4/9 → x = sin⁻¹(4/9)
- x ≈
26.4°
✔ Answer: 26.4°
---
2)
- Adjacent = 10 cm, Hypotenuse = 14 cm → Use
cos
- cos(x) = 10/14 → x = cos⁻¹(10/14)
- x ≈
44.4°
✔ Answer: 44.4°
---
3)
- Opposite = 7 cm, Adjacent = 12 cm → Use
tan
- tan(x) = 7/12 → x = tan⁻¹(7/12)
- x ≈
30.3°
✔ Answer: 30.3°
---
4)
- Adjacent = 15 cm, Hypotenuse = 18 cm → Use
cos
- cos(x) = 15/18 → x = cos⁻¹(15/18)
- x ≈
33.6°
✔ Answer: 33.6°
---
5)
- Opposite = 0.83 cm, Adjacent = 0.99 cm → Use
tan
- tan(x) = 0.83 / 0.99 ≈ 0.8384 → x = tan⁻¹(0.8384)
- x ≈
39.9°
✔ Answer: 39.9°
---
6)
- Opposite = 410 mm, Hypotenuse = 972 mm → Use
sin
- sin(x) = 410/972 → x = sin⁻¹(410/972)
- x ≈
24.9°
✔ Answer: 24.9°
---
## 🔷 SECTION B: Find the missing angles or lengths x
Some problems ask for an
angle, others for a
length. We apply SOH CAH TOA accordingly.
---
1)
- Opposite = 5 cm, Adjacent = 11 cm → Use
tan
- tan(x) = 5/11 → x = tan⁻¹(5/11)
- x ≈
24.4°
✔ Answer: 24.4°
---
2)
- Opposite = 0.21 cm, Hypotenuse = 0.7 cm → Use
sin
- sin(x) = 0.21 / 0.7 = 0.3 → x = sin⁻¹(0.3)
- x ≈
17.5°
✔ Answer: 17.5°
---
3)
- Opposite = 6.4 cm, Angle = 62° → Use
tan
- 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
✔ Answer: 3.40 cm
---
4)
- Opposite = 27 mm, Hypotenuse = 1.5 cm = 15 mm → Use
sin
- sin(x) = 27 / 15 = 1.8 → ❗Wait — this is impossible!
⚠️
Error Alert! The opposite side (27 mm) is longer than the hypotenuse (15 mm). This is
not possible in a right triangle.
➡️ Likely typo in question. If we assume the hypotenuse is 15 cm = 150 mm:
- sin(x) = 27 / 150 = 0.18 → x = sin⁻¹(0.18) ≈
10.4°
✔ Assuming typo corrected:
10.4°
*(Note: In real exam, flag this as impossible. But since worksheet expects answer, likely meant 150 mm or 2.7 cm.)*
---
5)
- Adjacent = 2.3 cm, Angle = 13° → Use
tan
- tan(13°) = Opposite / Adjacent = x / 2.3
- x = 2.3 × tan(13°)
- tan(13°) ≈ 0.2309 → x ≈ 2.3 × 0.2309 ≈
0.531 cm
✔ Answer: 0.531 cm
---
6)
- Opposite = 620 cm, Hypotenuse = 8 m = 800 cm → Use
sin
- sin(x) = 620 / 800 = 0.775 → x = sin⁻¹(0.775)
- x ≈
50.8°
✔ Answer: 50.8°
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##
✔ FINAL ANSWERS SUMMARY
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🟦 SECTION A
1)
26.4°
2)
44.4°
3)
30.3°
4)
33.6°
5)
39.9°
6)
24.9°
---
🟩 SECTION B
1)
24.4°
2)
17.5°
3)
3.40 cm
4)
10.4° *(assuming typo: 27 mm vs 150 mm hypotenuse)*
5)
0.531 cm
6)
50.8°
---
## 📝 Notes:
- Always check units (e.g., cm vs mm) — convert if needed.
- If sine or cosine > 1, it’s impossible — likely a typo.
- Use calculator in
degree mode.
- Round final answers to
3 significant figures.
Let me know if you’d like step-by-step diagrams or calculator keystrokes!
Parent Tip: Review the logic above to help your child master the concept of trigonometry ratios worksheet.
