Trigonometic Ratios (SOH CAH TOA): Missing Side Lengths Worksheet ... - Free Printable

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 — that tells us which ratio to use.

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Section A

#### 1) Right triangle, angle 35°, hypotenuse = 12 cm, find 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 (to 3 sig figs)

#### 2) Right triangle, angle 67°, hypotenuse = 9 cm, find 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

#### 3) Right triangle, angle 49°, adjacent = 16 cm, find 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

#### 4) Right triangle, angle 62°, opposite = 17 cm, find x (hypotenuse)
→ Use sin(62°) = opposite / hypotenuse = 17 / x
→ x = 17 / sin(62°)
→ sin(62°) ≈ 0.8829
→ x ≈ 17 / 0.8829 ≈ 19.254 → 19.3

#### 5) Right triangle, angle 27°, adjacent = 5 cm, find 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

#### 6) Right triangle, angle 33°, hypotenuse = 21 cm, find 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

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Section B

#### 1) Right triangle, angle 15°, adjacent = 14 cm, find 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

#### 2) Right triangle, angle 71°, opposite = 19 cm, find 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

#### 3) Right triangle, angle 42°, opposite = 5 cm, find 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

#### 4) Right triangle, angle 58°, adjacent = 11 cm, find 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

#### 5) Right triangle, angle 20°, opposite = 7 cm, find x (adjacent side)
→ Use tan(20°) = opposite / adjacent = 7 / x
→ x = 7 / tan(20°)
→ tan(20°) ≈ 0.3640
→ x ≈ 7 / 0.3640 ≈ 19.231 → 19.2

#### 6) Right triangle, angle 36°, opposite = 24 cm, find 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

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Final Answer:

Section A:
1) x = 6.88
2) x = 3.52
3) x = 18.4
4) x = 19.3
5) x = 2.55
6) x = 17.6

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