We are given a worksheet titled
“Using trigonometric ratios to find angles”, and we need to find the marked angles (θ, α, β, a, b, c, d, e, f) in each right-angled triangle using trigonometry. All answers should be correct to
2 decimal places.
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## 🔑 Key Trigonometric Ratios (SOHCAHTOA):
In a right-angled triangle:
-
sin(θ) = Opposite / Hypotenuse
-
cos(θ) = Adjacent / Hypotenuse
-
tan(θ) = Opposite / Adjacent
To find the angle θ, we use the
inverse functions:
- θ = arcsin(Opp/Hyp)
- θ = arccos(Adj/Hyp)
- θ = arctan(Opp/Adj)
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#
✔ Problem 1: Find θ
a)
Opposite = 4.8, Hypotenuse = 5.2
→ Use
sin(θ) = Opp/Hyp = 4.8 / 5.2
> θ = arcsin(4.8 / 5.2) ≈ arcsin(0.9230769) ≈
67.38°
✔ Answer: 67.38°
---
b)
Adjacent = 3.2, Hypotenuse = 4.7
→ Use
cos(θ) = Adj/Hyp = 3.2 / 4.7
> θ = arccos(3.2 / 4.7) ≈ arccos(0.680851) ≈
47.09°
✔ Answer: 47.09°
---
c)
Opposite = 8, Adjacent = 3
→ Use
tan(θ) = Opp/Adj = 8 / 3 ≈ 2.6667
> θ = arctan(8 / 3) ≈ arctan(2.6667) ≈
69.44°
✔ Answer: 69.44°
---
#
✔ Problem 2: Find θ or β
a)
Opposite = 17, Adjacent = 12
→ tan(θ) = 17 / 12 ≈ 1.4167
> θ = arctan(17/12) ≈
54.78°
✔ Answer: 54.78°
---
b)
This is a right triangle with hypotenuse = 7.2 m, adjacent to β = 4 m
→ cos(β) = Adj/Hyp = 4 / 7.2 ≈ 0.5556
> β = arccos(4 / 7.2) ≈
56.25°
✔ Answer: 56.25°
---
c)
Adjacent = 10, Hypotenuse = 12
→ cos(θ) = 10 / 12 ≈ 0.8333
> θ = arccos(10 / 12) ≈
33.56°
✔ Answer: 33.56°
---
#
✔ Problem 3: Find θ or α
a)
Opposite = ? Wait — hypotenuse = 8, adjacent = 2
→ cos(θ) = Adj/Hyp = 2 / 8 = 0.25
> θ = arccos(2 / 8) = arccos(0.25) ≈
75.52°
✔ Answer: 75.52°
---
b)
Hypotenuse = 5 m, adjacent to α = 3 m
→ cos(α) = 3 / 5 = 0.6
> α = arccos(0.6) ≈
53.13°
✔ Answer: 53.13°
---
c)
Opposite = 2.7, Adjacent = 3.5
→ tan(α) = 2.7 / 3.5 ≈ 0.7714
> α = arctan(2.7 / 3.5) ≈
37.62°
✔ Answer: 37.62°
---
#
✔ Problem 4: Find a, b, c, d, e, f
a)
Right triangle: opposite to angle
a = 13.5, adjacent = 15.3
→ tan(a) = 13.5 / 15.3 ≈ 0.8824
> a = arctan(13.5 / 15.3) ≈
41.41°
✔ Answer: 41.41°
---
b)
Right triangle: opposite to angle
c = 77.3, hypotenuse = 89.4
→ sin(c) = 77.3 / 89.4 ≈ 0.8647
> c = arcsin(77.3 / 89.4) ≈
59.85°
✔ Answer: 59.85°
---
c)
Right triangle: adjacent to angle
b = 92.7, hypotenuse = 106.4
→ cos(b) = 92.7 / 106.4 ≈ 0.8713
> b = arccos(92.7 / 106.4) ≈
29.38°
✔ Answer: 29.38°
---
d)
Right triangle: opposite to angle
d = 18.7, hypotenuse = 43.7
→ sin(d) = 18.7 / 43.7 ≈ 0.4279
> d = arcsin(18.7 / 43.7) ≈
25.34°
✔ Answer: 25.34°
---
e)
Right triangle: adjacent to angle
e = 13.85, hypotenuse = 18.56
→ cos(e) = 13.85 / 18.56 ≈ 0.7462
> e = arccos(13.85 / 18.56) ≈
41.73°
✔ Answer: 41.73°
---
f)
Right triangle: adjacent to angle
α = 9.8 cm, hypotenuse = 12.2 cm
→ cos(α) = 9.8 / 12.2 ≈ 0.8033
> α = arccos(9.8 / 12.2) ≈
36.54°
✔ Answer: 36.54°
---
# 📋 Final Answers:
## Problem 1:
a)
67.38°
b)
47.09°
c)
69.44°
## Problem 2:
a)
54.78°
b)
56.25°
c)
33.56°
## Problem 3:
a)
75.52°
b)
53.13°
c)
37.62°
## Problem 4:
a)
41.41°
b)
59.85°
c)
29.38°
d)
25.34°
e)
41.73°
f)
36.54°
---
✔ All answers are rounded to
2 decimal places as required.
Let me know if you’d like step-by-step calculator instructions or diagrams!
Parent Tip: Review the logic above to help your child master the concept of sohcahtoa word problems worksheet.
