Trigonometry Worksheet T4 – Calculating Angles

Let’s solve each problem step by step using trigonometry (SOHCAHTOA).

We’ll use:
- Sine = Opposite / Hypotenuse → angle = sin⁻¹(opposite/hypotenuse)
- Cosine = Adjacent / Hypotenuse → angle = cos⁻¹(adjacent/hypotenuse)
- Tangent = Opposite / Adjacent → angle = tan⁻¹(opposite/adjacent)

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Problem 1:
Triangle with hypotenuse = 10 cm, opposite side to angle s = 4 cm.
→ Use sine: sin(s) = 4/10 = 0.4
→ s = sin⁻¹(0.4) ≈ 23.6°

✔ Check: sin(23.6°) ≈ 0.4 — correct.

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Problem 2:
Adjacent to angle c = 6 cm, hypotenuse = 12 cm.
→ Use cosine: cos(c) = 6/12 = 0.5
→ c = cos⁻¹(0.5) = 60°

✔ Check: cos(60°) = 0.5 — correct.

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Problem 3:
Opposite to angle t = 18 cm, adjacent = 9 cm.
→ Use tangent: tan(t) = 18/9 = 2
→ t = tan⁻¹(2) ≈ 63.4°

✔ Check: tan(63.4°) ≈ 2 — correct.

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Problem 4:
Angle w is between sides 20 cm and 19 cm? Wait — let’s look again.

Actually, in triangle 4, the right angle is at the top left. Angle w is at the bottom left. The side opposite to w is 19 cm, and the hypotenuse is 20 cm.

Wait — no: if right angle is at top left, then for angle w (bottom left), the adjacent side is 20 cm? That can’t be — hypotenuse must be longest side.

Looking carefully: sides are 19 cm and 20 cm, and right angle is shown at the vertex between them? No — actually, the right angle mark is at the top-left corner. So the two legs are unknown? Wait — labels say “19cm” along one leg, “20cm” along the other? But that would make hypotenuse √(19² + 20²) ≈ 27.6 — but it’s not labeled.

Wait — re-examining: In problem 4, the triangle has a right angle at the top-left vertex. Side from top-left to bottom-right is labeled 19 cm? Or is 19 cm the side opposite angle w?

Actually, looking at standard labeling: angle w is at bottom-left. The side opposite to w is the vertical side (which should be 19 cm?), and the hypotenuse is the side opposite the right angle — which is the long diagonal, labeled 20 cm.

Yes! So:

For angle w:
- Opposite = 19 cm
- Hypotenuse = 20 cm
→ sin(w) = 19/20 = 0.95
→ w = sin⁻¹(0.95) ≈ 71.8°

✔ Check: sin(71.8°) ≈ 0.95 — correct.

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Problem 5:
Right angle at top-left. Angle x is at top-right. Adjacent side to x is 11 cm, hypotenuse is 23 cm.

→ cos(x) = adjacent/hypotenuse = 11/23 ≈ 0.4783
→ x = cos⁻¹(0.4783) ≈ 61.4°

✔ Check: cos(61.4°) ≈ 0.478 — correct.

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Problem 6:
Right angle at top-right. Angle b is at bottom-right. Adjacent to b is the vertical side (unknown), but we have:

Hypotenuse = 8 cm, adjacent to b = ? Wait — side labeled 5 cm is horizontal, from top-left to top-right. Since right angle is at top-right, then:

- Horizontal leg = 5 cm (this is adjacent to angle b)
- Hypotenuse = 8 cm

So cos(b) = adjacent/hypotenuse = 5/8 = 0.625
→ b = cos⁻¹(0.625) ≈ 51.3°

✔ Check: cos(51.3°) ≈ 0.625 — correct.

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Problem 7:
Right angle at top-right. Angle z is at top-left. Side adjacent to z is 7 cm, hypotenuse is 24 cm? Wait — no.

Sides: 7 cm and 24 cm are both legs? Because right angle is at top-right, so the two legs are 7 cm (top) and 24 cm (right side)? Then hypotenuse is the long side.

But angle z is at top-left. For angle z:
- Opposite side = 24 cm (vertical leg)
- Adjacent side = 7 cm (horizontal leg)

→ tan(z) = opposite/adjacent = 24/7 ≈ 3.4286
→ z = tan⁻¹(3.4286) ≈ 73.7°

✔ Check: tan(73.7°) ≈ 3.43 — correct.

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Problem 8:
Right angle at top-left. Angle y is at top-right. Adjacent to y is 17 cm, opposite is 13 cm.

→ tan(y) = opposite/adjacent = 13/17 ≈ 0.7647
→ y = tan⁻¹(0.7647) ≈ 37.4°

✔ Check: tan(37.4°) ≈ 0.765 — correct.

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Problem 9:
Right angle at bottom-right. Angle v is at top-left. Opposite to v is 21 cm, hypotenuse is 22 cm.

→ sin(v) = opposite/hypotenuse = 21/22 ≈ 0.9545
→ v = sin⁻¹(0.9545) ≈ 72.6°

✔ Check: sin(72.6°) ≈ 0.954 — correct.

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Problem 10:
This is an isosceles right triangle? Right angle at top. Two equal legs: 15 cm and 16 cm? Wait — they’re not equal. Actually, legs are 15 cm and 16 cm, hypotenuse is base.

Angle u is at bottom-left. Adjacent to u is 15 cm? Let’s see:

From bottom-left corner (angle u):
- Adjacent side = 15 cm (left leg)
- Opposite side = 16 cm (right leg)

→ tan(u) = opposite/adjacent = 16/15 ≈ 1.0667
→ u = tan⁻¹(1.0667) ≈ 46.8°

✔ Check: tan(46.8°) ≈ 1.067 — correct.

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Final Answers:

1. 23.6°
2. 60°
3. 63.4°
4. 71.8°
5. 61.4°
6. 51.3°
7. 73.7°
8. 37.4°
9. 72.6°
10. 46.8°

All rounded to one decimal place as appropriate.

Final Answer:
1. 23.6°, 2. 60°, 3. 63.4°, 4. 71.8°, 5. 61.4°, 6. 51.3°, 7. 73.7°, 8. 37.4°, 9. 72.6°, 10. 46.8°