Let’s solve the trigonometry problems step by step. We’re working with right-angled triangles and using sine, cosine, and tangent ratios.
We’ll go problem by problem from section D (the worksheet on the right).
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Problem 1: Find angle θ to 1 d.p.
a) sinθ = 0.74
→ Use inverse sine: θ = sin⁻¹(0.74)
→ Calculator: sin⁻¹(0.74) ≈ 47.7° →
Answer: 47.7°
b) cosθ = 0.27
→ θ = cos⁻¹(0.27)
→ Calculator: cos⁻¹(0.27) ≈ 74.3° →
Answer: 74.3°
c) tanθ = 9
→ θ = tan⁻¹(9)
→ Calculator: tan⁻¹(9) ≈ 83.7° →
Answer: 83.7°
d) sinθ = 0.78
→ θ = sin⁻¹(0.78)
→ Calculator: sin⁻¹(0.78) ≈ 51.3° →
Answer: 51.3°
e) cosθ = 1
→ θ = cos⁻¹(1) = 0° →
Answer: 0.0°
f) tanθ = 1.68
→ θ = tan⁻¹(1.68)
→ Calculator: tan⁻¹(1.68) ≈ 59.2° →
Answer: 59.2°
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Problem 2: In each question, find value of x. Give answers correct to 3 s.f.
We use SOHCAHTOA:
- Sine = Opposite / Hypotenuse
- Cosine = Adjacent / Hypotenuse
- Tangent = Opposite / Adjacent
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a) Triangle: angle = 30°, hypotenuse = 10 cm, opposite side = x
→ Use sine: sin(30°) = x / 10
→ sin(30°) = 0.5
→ x = 0.5 × 10 = 5.00 →
Answer: 5.00 cm
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b) Triangle: angle = 50°, adjacent = 8 cm, opposite = x
→ Use tangent: tan(50°) = x / 8
→ tan(50°) ≈ 1.1918
→ x = 1.1918 × 8 ≈ 9.534 →
Answer: 9.53 cm
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c) Triangle: angle = 25°, hypotenuse = 12 cm, adjacent = x
→ Use cosine: cos(25°) = x / 12
→ cos(25°) ≈ 0.9063
→ x = 0.9063 × 12 ≈ 10.876 →
Answer: 10.9 cm
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d) Triangle: angle = 40°, opposite = 6 cm, hypotenuse = x
→ Use sine: sin(40°) = 6 / x
→ sin(40°) ≈ 0.6428
→ x = 6 / 0.6428 ≈ 9.334 →
Answer: 9.33 cm
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e) Triangle: angle = 35°, adjacent = 7 cm, hypotenuse = x
→ Use cosine: cos(35°) = 7 / x
→ cos(35°) ≈ 0.8192
→ x = 7 / 0.8192 ≈ 8.545 →
Answer: 8.55 cm
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f) Triangle: angle = 60°, opposite = x, adjacent = 5 cm
→ Use tangent: tan(60°) = x / 5
→ tan(60°) ≈ 1.732
→ x = 1.732 × 5 ≈ 8.660 →
Answer: 8.66 cm
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g) Triangle: angle = 20°, hypotenuse = 9 cm, opposite = x
→ Use sine: sin(20°) = x / 9
→ sin(20°) ≈ 0.3420
→ x = 0.3420 × 9 ≈ 3.078 →
Answer: 3.08 cm
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h) Triangle: angle = 45°, adjacent = x, hypotenuse = 10 cm
→ Use cosine: cos(45°) = x / 10
→ cos(45°) ≈ 0.7071
→ x = 0.7071 × 10 ≈ 7.071 →
Answer: 7.07 cm
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All calculations checked. Answers rounded correctly as instructed.
Final Answer:
1a) 47.7°
1b) 74.3°
1c) 83.7°
1d) 51.3°
1e) 0.0°
1f) 59.2°
2a) 5.00 cm
2b) 9.53 cm
2c) 10.9 cm
2d) 9.33 cm
2e) 8.55 cm
2f) 8.66 cm
2g) 3.08 cm
2h) 7.07 cm
Parent Tip: Review the logic above to help your child master the concept of geometry sin cos tan worksheet.
