Mixed Practice: Trig Ratios + Pythagorean for Side Lengths

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| Questions: 20 | Updated: Nov 10, 2025
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1) In a right triangle, angle A = 30°. The side adjacent to A is 5. Find the hypotenuse.

Explanation

cos 30° = adjacent / hypotenuse = 5 / H; H = 5 / (√3/2) = 10/√3 = (10√3)/3. Hence, hypotenuse = 10√3/3.

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About This Quiz
Mixed Practice: Trig Ratios + Pythagorean For Side Lengths - Quiz

Think you can juggle multiple methods? In this quiz, you’ll mix SOHCAHTOA with the Pythagorean Theorem to solve multi-step problems. You’ll move between ratios and square-sum checks, switch forms (like from sin to cos), and use inverse trig when needed. Expect problems that require planning, tidy arithmetic, and smart verification.

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2) Sin B = 3/5. Find the length of the adjacent side if the hypotenuse is 20.

Explanation

cos B = √(1 − sin²B) = √(1 − (3/5)²) = 4/5; adjacent = 20 × (4/5) = 16. Hence, adjacent = 16.

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3) A triangle has tan θ = 4/3. If the opposite side is 8, find the adjacent side and hypotenuse.

Explanation

tan θ = opposite / adjacent = 4/3; adjacent = 8 ÷ (4/3) = 6; hypotenuse = √(8² + 6²) = 10.

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4) Adjacent = 9, angle = 60°. Find the hypotenuse.

Explanation

cos 60° = adjacent / hypotenuse = 9 / H; H = 9 / 0.5 = 18. Hence, hypotenuse = 18.

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5) If tan θ = 5/12 and the hypotenuse = 13, find the adjacent side.

Explanation

Right triangle 5–12–13 ⇒ opposite:adjacent:hypotenuse = 5:12:13. Given hypotenuse = 13 ⇒ adjacent = 12.

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6) In a right triangle, cos θ = 0.8 and the hypotenuse = 25. Find the opposite side.

Explanation

adjacent = 25 × 0.8 = 20; opposite = √(25² − 20²) = 15. Hence, opposite = 15.

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7) First find the hypotenuse, then the missing side: adjacent = 9, opposite = 12.

Explanation

Hypotenuse = √(9² + 12²) = 15; sin θ = opposite / hypotenuse = 12 / 15.

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8) If sin α = 4/5 and the opposite = 24, find the hypotenuse and adjacent.

Explanation

hypotenuse = 24 / (4/5) = 30; cos α = √(1 − (4/5)²) = 3/5 ⇒ adjacent = 18.

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9) Given tan β = 1.5 and adjacent = 6, find the opposite and hypotenuse (nearest tenth).

Explanation

opposite = 6 × 1.5 = 9.0; hypotenuse = √(6² + 9²) ≈ 10.8.

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10) In a 45°–45°–90° triangle with leg = 7, find the hypotenuse.

Explanation

hypotenuse = leg × √2 = 7√2.

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11) Cos θ = 0.6. Find tan θ.

Explanation

sin θ = √(1 − 0.36) = 0.8; tan θ = 0.8 / 0.6 = 1.33 (not listed).

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12) Opposite = 9, adjacent = 40. Find the hypotenuse and sin θ.

Explanation

hypotenuse = √(9² + 40²) = 41; sin θ = 9 / 41.

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13) If sin x = 5/13 and the hypotenuse = 26, find cos x and adjacent.

Explanation

cos x = √(1 − (5/13)²) = 12/13; adjacent = 26 × (12/13) = 24.

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14) Angle θ = 35°, adjacent = 12. Find the opposite side (nearest tenth).

Explanation

opposite = 12 × tan 35° ≈ 8.4.

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15) Tan θ = 3/4, hypotenuse = 25. Find opposite and adjacent.

Explanation

Scale 3–4–5 by k with 5k = 25 ⇒ k = 5 ⇒ opposite = 15, adjacent = 20.

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16) Sin α = 0.5 and cos α = √3 / 2. Find tan α.

Explanation

tan α = (1/2) / (√3/2) = 1/√3.

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17) If opposite = 10 and tan θ = 2. Find the adjacent and hypotenuse (nearest tenth).

Explanation

tan θ = 2 ⇒ adjacent = 10/2 = 5; hypotenuse = √(10² + 5²) ≈ 11.2.

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18) Angle θ = 70°, hypotenuse = 40. Find opposite (nearest tenth).

Explanation

opposite = 40 × sin 70° ≈ 37.6.

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19) Sin θ = 24/25. Find cos θ and tan θ.

Explanation

cos θ = √(1 − (24/25)²) = 7/25; tan θ = (24/25)/(7/25) = 24/7.

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20) If adjacent = 9 and θ = 40°, first find opposite, then find hypotenuse.

Explanation

opposite = 9 × tan 40° ≈ 7.6; hypotenuse = 9 / cos 40° ≈ 11.8.

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In a right triangle, angle A = 30°. The side adjacent to A is 5....
Sin B = 3/5. Find the length of the adjacent side if the hypotenuse is...
A triangle has tan θ = 4/3. If the opposite side is 8, find the...
Adjacent = 9, angle = 60°. Find the hypotenuse.
If tan θ = 5/12 and the hypotenuse = 13, find the adjacent side.
In a right triangle, cos θ = 0.8 and the hypotenuse = 25. Find the...
First find the hypotenuse, then the missing side: adjacent = 9,...
If sin α = 4/5 and the opposite = 24, find the hypotenuse and...
Given tan β = 1.5 and adjacent = 6, find the opposite and hypotenuse...
In a 45°–45°–90° triangle with leg = 7, find the hypotenuse.
Cos θ = 0.6. Find tan θ.
Opposite = 9, adjacent = 40. Find the hypotenuse and sin θ.
If sin x = 5/13 and the hypotenuse = 26, find cos x and adjacent.
Angle θ = 35°, adjacent = 12. Find the opposite side (nearest...
Tan θ = 3/4, hypotenuse = 25. Find opposite and adjacent.
Sin α = 0.5 and cos α = √3 / 2. Find tan α.
If opposite = 10 and tan θ = 2. Find the adjacent and hypotenuse...
Angle θ = 70°, hypotenuse = 40. Find opposite (nearest tenth).
Sin θ = 24/25. Find cos θ and tan θ.
If adjacent = 9 and θ = 40°, first find opposite, then find...
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