Trigonometry (A) Missing Lengths worksheet featuring 12 problems with right triangles and angles, requiring calculation of unknown sides.

Let’s solve each problem one by one. We’ll use trigonometry ratios:
- Sine (sin) = opposite / hypotenuse
- Cosine (cos) = adjacent / hypotenuse
- Tangent (tan) = opposite / adjacent

We always look at the angle given, then decide which sides we have and which side is missing (x). Then pick the right ratio.

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Section A

#### 1) Triangle with angle 35°, hypotenuse = 12 cm, x = opposite side
→ Use sin(35°) = opposite / hypotenuse = x / 12
→ x = 12 × sin(35°)
→ sin(35°) ≈ 0.5736
→ x ≈ 12 × 0.5736 = 6.8832 → 6.88 cm (3 sig figs)

#### 2) Angle 67°, hypotenuse = 9 cm, x = adjacent side
→ Use cos(67°) = adjacent / hypotenuse = x / 9
→ x = 9 × cos(67°)
→ cos(67°) ≈ 0.3907
→ x ≈ 9 × 0.3907 = 3.5163 → 3.52 cm

#### 3) Angle 49°, adjacent = 16 cm, x = opposite side
→ Use tan(49°) = opposite / adjacent = x / 16
→ x = 16 × tan(49°)
→ tan(49°) ≈ 1.1504
→ x ≈ 16 × 1.1504 = 18.4064 → 18.4 cm

#### 4) Angle 62°, adjacent = 17 cm, x = hypotenuse
→ Use cos(62°) = adjacent / hypotenuse = 17 / x
→ So, x = 17 / cos(62°)
→ cos(62°) ≈ 0.4695
→ x ≈ 17 / 0.4695 ≈ 36.208 → 36.2 cm

#### 5) Angle 27°, adjacent = 5 cm, x = opposite side
→ Use tan(27°) = opposite / adjacent = x / 5
→ x = 5 × tan(27°)
→ tan(27°) ≈ 0.5095
→ x ≈ 5 × 0.5095 = 2.5475 → 2.55 cm

#### 6) Angle 33°, hypotenuse = 21 cm, x = adjacent side
→ Use cos(33°) = adjacent / hypotenuse = x / 21
→ x = 21 × cos(33°)
→ cos(33°) ≈ 0.8387
→ x ≈ 21 × 0.8387 = 17.6127 → 17.6 cm

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Section B

#### 1) Angle 15°, adjacent = 14 cm, x = hypotenuse
→ Use cos(15°) = adjacent / hypotenuse = 14 / x
→ x = 14 / cos(15°)
→ cos(15°) ≈ 0.9659
→ x ≈ 14 / 0.9659 ≈ 14.494 → 14.5 cm

#### 2) Angle 71°, opposite = 19 cm, x = hypotenuse
→ Use sin(71°) = opposite / hypotenuse = 19 / x
→ x = 19 / sin(71°)
→ sin(71°) ≈ 0.9455
→ x ≈ 19 / 0.9455 ≈ 20.095 → 20.1 cm

#### 3) Angle 42°, opposite = 5 cm, x = adjacent side
→ Use tan(42°) = opposite / adjacent = 5 / x
→ x = 5 / tan(42°)
→ tan(42°) ≈ 0.9004
→ x ≈ 5 / 0.9004 ≈ 5.553 → 5.55 cm

#### 4) Angle 58°, adjacent = 11 cm, x = hypotenuse
→ Use cos(58°) = adjacent / hypotenuse = 11 / x
→ x = 11 / cos(58°)
→ cos(58°) ≈ 0.5299
→ x ≈ 11 / 0.5299 ≈ 20.758 → 20.8 cm

#### 5) Angle 20°, opposite = 7 cm, x = adjacent side
→ Use tan(20°) = opposite / adjacent = 7 / x
→ x = 7 / tan(20°)
→ tan(20°) ≈ 0.3640
→ x ≈ 7 / 0.3640 ≈ 19.230 → 19.2 cm

#### 6) Angle 36°, opposite = 24 cm, x = hypotenuse
→ Use sin(36°) = opposite / hypotenuse = 24 / x
→ x = 24 / sin(36°)
→ sin(36°) ≈ 0.5878
→ x ≈ 24 / 0.5878 ≈ 40.830 → 40.8 cm

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Final Answer:

Section A:
1) x = 6.88 cm
2) x = 3.52 cm
3) x = 18.4 cm
4) x = 36.2 cm
5) x = 2.55 cm
6) x = 17.6 cm

Section B:
1) x = 14.5 cm
2) x = 20.1 cm
3) x = 5.55 cm
4) x = 20.8 cm
5) x = 19.2 cm
6) x = 40.8 cm