SOHCAHTOA: Identify Ratios & Use Pythagoras in Right Triangles

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Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.
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| Attempts: 17 | Questions: 20 | Updated: Jan 22, 2026
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1) In a right triangle, the opposite side to angle θ is 9 and the hypotenuse is 15. What is sin θ?

Explanation

sin θ = opposite / hypotenuse

= 9 / 15

= 3 / 5.

Hence, sin θ = 3/5.

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About This Quiz
Sohcahtoa: Identify Ratios & Use Pythagoras In Right Triangles - Quiz

Develop a solid understanding of the trigonometric ratios sine, cosine, and tangent using the SOHCAHTOA method. This quiz focuses on identifying sides relative to an angle, computing missing sides using ratios, and applying the Pythagorean Theorem. Students practice finding exact trigonometric values, solving for unknown sides, and relating triangle geometry... see moreto real-world examples like ladders, ramps, and shadows. It builds fluency in interpreting and applying sine, cosine, and tangent relationships in right triangles.
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2) In a right triangle, the adjacent side to angle α is 7 and the hypotenuse is 25. Find cos α.

Explanation

cos α = adjacent / hypotenuse

= 7 / 25.

Hence, cos α = 7/25.

Submit

3) A right triangle has hypotenuse 50 and one leg 14. What is the other leg?

Explanation

Other leg² = 50² − 14²

= 2500 − 196

= 2304 ⇒ other leg = √2304 = 48.

Hence, the other leg is 48.

Submit

4) A right triangle has legs of 8 and 15. What is the hypotenuse?

Explanation

c² = 8² + 15²

= 64 + 225

= 289 ⇒ c = √289 = 17.

Hence, the hypotenuse is 17.

Submit

5) A 26 ft ladder leans against a wall. The base is 10 ft from the wall. How high up the wall does it reach?

Explanation

height² = 26² − 10²

= 676 − 100

= 576 ⇒ height = √576 = 24 ft.

Hence, the ladder reaches 24 ft.

Submit

6) If tan β = 5/12, what is the ratio of opposite to adjacent?

Explanation

tan β = opposite / adjacent

= 5 / 12.

Hence, opposite : adjacent = 5 : 12 = 5/12.

Submit

7) A right triangle has opposite = 14 and adjacent = 48 relative to angle θ. What is tan θ?

Explanation

tan θ = opposite / adjacent

= 14 / 48

= 7 / 24 (divide by 2).

Hence, tan θ = 7/24.

Submit

8) In a right triangle, adjacent = 11 and opposite = 60. What is the hypotenuse?

Explanation

c² = 11² + 60²

= 121 + 3600

= 3721 ⇒ c = √3721 = 61.

Hence, the hypotenuse is 61.

Submit

9) A ramp rises 2 ft and has a run of 24 ft. Find sin θ.

Explanation

Hypotenuse = √(24² + 2²)

= √(576 + 4)

= √580.

sin θ = opposite / hypotenuse

= 2 / √580

= 2 / √(24² + 2²).

Hence, sin θ = 2 / √(24² + 2²).

Submit

10) In a right triangle, the hypotenuse is 13 and one leg is 12. What is the other leg?

Explanation

Other leg² = 13² − 12²

= 169 − 144

= 25 ⇒ other leg = √25 = 5.

Hence, the other leg is 5.

Submit

11) If cos γ = 3/5 and the hypotenuse is 20, what is the adjacent side?

Explanation

cos γ = adjacent / hypotenuse = 3/5

adjacent = (3/5) × 20

= 12.

Hence, adjacent = 12.

Submit

12) In the same triangle as question 13, what is cos A?

Explanation

cos A = adjacent / hypotenuse

= 12 / 15

= 4 / 5.

Hence, cos A = 4/5.

Submit

13) A tree casts a 30 ft shadow. If tan θ = 4/3 for the sun’s angle, how tall is the tree?

Explanation

tan θ = height / shadow

= height / 30 = 4 / 3.

height = (4/3) × 30

= 40 ft.

Hence, the tree’s height is 40 ft.

Submit

14) A kite string is 65 m long. The kite is 33 m high. Assuming a straight string, what is sin θ?

Explanation

sin θ = opposite / hypotenuse

= 33 / 65.

Hence, sin θ = 33/65.

Submit

15) In a right triangle, sin α = 8/17. What is cos α?

Explanation

cos α = √(1 − sin² α)

= √(1 − (8/17)²)

= √(1 − 64/289)

= √(225/289).

Hence, cos α = 15/17.

Submit

16) A road climbs 9 m for every 120 m of horizontal distance. What is tan θ?

Explanation

tan θ = opposite / adjacent

= 9 / 120

= 3 / 40 (divide by 3).

Hence, tan θ = 3/40.

Submit

17) A right triangle has legs 9 cm and 12 cm. What is sin θ where θ is opposite the 12 cm side?

Explanation

Hypotenuse = √(9² + 12²)

= √(81 + 144)

= √225

= 15.

sin θ = opposite / hypotenuse

= 12 / 15

= 4 / 5.

Hence, sin θ = 4/5.

Submit

18) In right triangle ABC, angle A is opposite side BC = 9, adjacent side AC = 12, and hypotenuse AB = 15. What is sin A?

Explanation

sin A = opposite / hypotenuse

= 9 / 15

= 3 / 5.

Hence, sin A = 3/5.

Submit

19) From the top of a 40 m building, the angle of depression to a car has tan θ = 40/30. How far is the car from the base?

Explanation

tan θ = opposite / adjacent

= 40 / x and also = 40 / 30.

40 / x = 40 / 30 ⇒ x = 30.

Hence, the car is 30 m from the base.

Submit

20) In a right triangle, cos β = 5/13 and the adjacent side is 15. What is the hypotenuse?

Explanation

cos β = adjacent / hypotenuse = 5 / 13

hypotenuse = adjacent ÷ (5/13)

= 15 × (13/5)

= 39.

Hence, the hypotenuse is 39.

Submit
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Cierra Henderson |MBA |
K-12 Expert
Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.
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In a right triangle, the opposite side to angle θ is 9 and the...
In a right triangle, the adjacent side to angle α is 7 and the...
A right triangle has hypotenuse 50 and one leg 14. What is the other...
A right triangle has legs of 8 and 15. What is the hypotenuse?
A 26 ft ladder leans against a wall. The base is 10 ft from the wall....
If tan β = 5/12, what is the ratio of opposite to adjacent?
A right triangle has opposite = 14 and adjacent = 48 relative to angle...
In a right triangle, adjacent = 11 and opposite = 60. What is the...
A ramp rises 2 ft and has a run of 24 ft. Find sin θ.
In a right triangle, the hypotenuse is 13 and one leg is 12. What is...
If cos γ = 3/5 and the hypotenuse is 20, what is the adjacent side?
In the same triangle as question 13, what is cos A?
A tree casts a 30 ft shadow. If tan θ = 4/3 for the sun’s angle,...
A kite string is 65 m long. The kite is 33 m high. Assuming a straight...
In a right triangle, sin α = 8/17. What is cos α?
A road climbs 9 m for every 120 m of horizontal distance. What is tan...
A right triangle has legs 9 cm and 12 cm. What is sin θ where θ is...
In right triangle ABC, angle A is opposite side BC = 9, adjacent side...
From the top of a 40 m building, the angle of depression to a car has...
In a right triangle, cos β = 5/13 and the adjacent side is 15. What...
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