Let’s solve each triangle one by one. We’re told
not to use Pythagoras, so we’ll use trigonometry: sine, cosine, and tangent.
Remember for a right-angled triangle:
- sin(θ) = opposite / hypotenuse
- cos(θ) = adjacent / hypotenuse
- tan(θ) = opposite / adjacent
We’ll label the sides relative to the given angle (except the right angle).
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Triangle 1:
Right angle at B. Angle at A is 30°. Hypotenuse AC = 6 cm.
We need to find AB (adjacent to 30°) and BC (opposite to 30°).
→ Use cos(30°) = adjacent / hypotenuse = AB / 6
AB = 6 × cos(30°) ≈ 6 × 0.8660 ≈
5.196 cm
→ Use sin(30°) = opposite / hypotenuse = BC / 6
BC = 6 × sin(30°) = 6 × 0.5 =
3 cm
✔ Check: Doesn’t need Pythagoras — we used trig only.
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Triangle 2:
Right angle at B. Angle at A is 20°. Hypotenuse AC = 9 cm.
Find AB (adjacent) and BC (opposite).
→ cos(20°) = AB / 9 → AB = 9 × cos(20°) ≈ 9 × 0.9397 ≈
8.457 cm
→ sin(20°) = BC / 9 → BC = 9 × sin(20°) ≈ 9 × 0.3420 ≈
3.078 cm
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Triangle 3:
Right angle at B. Angle at C is 25°. Side CB = 12 cm (this is adjacent to angle C).
We need to find AB (opposite to angle C) and AC (hypotenuse).
→ tan(25°) = opposite / adjacent = AB / 12
AB = 12 × tan(25°) ≈ 12 × 0.4663 ≈
5.596 cm
→ cos(25°) = adjacent / hypotenuse = 12 / AC
AC = 12 / cos(25°) ≈ 12 / 0.9063 ≈
13.241 cm
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Triangle 4:
Right angle at B. Angle at C is 40°. Side CB = 10 cm (adjacent to angle C).
Find AB (opposite) and AC (hypotenuse).
→ tan(40°) = AB / 10 → AB = 10 × tan(40°) ≈ 10 × 0.8391 ≈
8.391 cm
→ cos(40°) = 10 / AC → AC = 10 / cos(40°) ≈ 10 / 0.7660 ≈
13.055 cm
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Triangle 5:
Right angle at B. Angle at A is 30°. Side AB = 2 cm (adjacent to angle A).
Find BC (opposite) and AC (hypotenuse).
→ tan(30°) = BC / 2 → BC = 2 × tan(30°) ≈ 2 × 0.5774 ≈
1.155 cm
→ cos(30°) = 2 / AC → AC = 2 / cos(30°) ≈ 2 / 0.8660 ≈
2.309 cm
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Triangle 6:
Right angle at B. Angle at A is 45°. Side AB = 3 cm (adjacent to angle A).
Since it’s 45°, this is an isosceles right triangle → opposite = adjacent.
So BC = AB =
3 cm
Hypotenuse AC = ?
→ cos(45°) = 3 / AC → AC = 3 / cos(45°) ≈ 3 / 0.7071 ≈
4.243 cm
Alternatively, since both legs are 3, you could do √(3² + 3²) but that’s Pythagoras — not allowed! So stick with trig.
cos(45°) = adjacent/hypotenuse → same as above.
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Now let’s write all answers clearly.
Final Answer:
1. AB ≈ 5.20 cm, BC = 3.00 cm
2. AB ≈ 8.46 cm, BC ≈ 3.08 cm
3. AB ≈ 5.60 cm, AC ≈ 13.24 cm
4. AB ≈ 8.39 cm, AC ≈ 13.06 cm
5. BC ≈ 1.16 cm, AC ≈ 2.31 cm
6. BC = 3.00 cm, AC ≈ 4.24 cm
(Rounded to 2 decimal places unless exact)
Parent Tip: Review the logic above to help your child master the concept of trig worksheet with answers.
