To find the missing side lengths in these right triangles, we use trigonometric ratios. The three main ratios are:
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Sine (sin) = opposite / hypotenuse
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Cosine (cos) = adjacent / hypotenuse
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Tangent (tan) = opposite / adjacent
We pick the ratio that matches the sides we know and the side we’re solving for.
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Problem 1:
Given: angle = 48°, hypotenuse = 12, find adjacent side (x)
→ Use cosine: cos(48°) = adjacent / hypotenuse = x / 12
→ x = 12 × cos(48°)
→ cos(48°) ≈ 0.6691
→ x ≈ 12 × 0.6691 ≈ 8.03 →
rounded to nearest tenth: 8.0
Wait — let me double-check with calculator precision.
Actually, using more precise value:
cos(48°) = 0.669130606...
12 × 0.669130606 ≈ 8.029567 → rounds to
8.0
But wait — looking at the diagram again: is “x” the adjacent or opposite?
In problem 1: triangle has right angle at bottom left, 48° at bottom right, so side labeled “x” is the base — which is adjacent to the 48° angle. Hypotenuse is 12. So yes, cosine is correct.
✔ Problem 1 answer:
8.0
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Problem 2:
Given: angle = 40°, opposite side = 15, find hypotenuse (x)
→ Use sine: sin(40°) = opposite / hypotenuse = 15 / x
→ x = 15 / sin(40°)
→ sin(40°) ≈ 0.6428
→ x ≈ 15 / 0.6428 ≈ 23.335 →
rounded to nearest tenth: 23.3
Check: 15 ÷ sin(40°) = 15 ÷ 0.6427876097 ≈ 23.335 → yes, 23.3
✔ Problem 2 answer:
23.3
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Problem 3:
Given: angle = 55°, adjacent side = 9, find opposite side (x)
→ Use tangent: tan(55°) = opposite / adjacent = x / 9
→ x = 9 × tan(55°)
→ tan(55°) ≈ 1.4281
→ x ≈ 9 × 1.4281 ≈ 12.853 →
rounded to nearest tenth: 12.9
Check: tan(55°) = 1.428148... × 9 = 12.8533 → rounds to 12.9
✔ Problem 3 answer:
12.9
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Problem 4:
Given: angle = 30°, hypotenuse = 14, find opposite side (x)
→ Use sine: sin(30°) = opposite / hypotenuse = x / 14
→ sin(30°) = 0.5 exactly
→ x = 14 × 0.5 = 7.0
✔ Problem 4 answer:
7.0
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Problem 5:
Given: angle = 40°, hypotenuse = 12, find opposite side (x)
→ Use sine: sin(40°) = x / 12
→ x = 12 × sin(40°)
→ sin(40°) ≈ 0.6428
→ x ≈ 12 × 0.6428 ≈ 7.7136 →
rounded to nearest tenth: 7.7
Check: 12 × 0.6427876097 ≈ 7.71345 → rounds to 7.7
✔ Problem 5 answer:
7.7
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Problem 6:
Given: angle = 18°, adjacent side = 20, find opposite side (x)
→ Use tangent: tan(18°) = x / 20
→ x = 20 × tan(18°)
→ tan(18°) ≈ 0.3249
→ x ≈ 20 × 0.3249 ≈ 6.498 →
rounded to nearest tenth: 6.5
Check: tan(18°) = 0.324919696... × 20 = 6.49839 → rounds to 6.5
✔ Problem 6 answer:
6.5
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Final Answer:
1. 8.0
2. 23.3
3. 12.9
4. 7.0
5. 7.7
6. 6.5
Parent Tip: Review the logic above to help your child master the concept of solve right triangles worksheet.
