Apply SOHCAHTOA: Solve for Unknown Sides & Angles

  • 10th Grade
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Cierra Henderson, MBA |
K-12 Expert
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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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| Questions: 20 | Updated: Jan 22, 2026
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1) In right triangle ABC with right angle at C, if sin(A) = 3/5, what is cos(A)?

Explanation

sin(A) = 3/5 ⇒ opposite : hypotenuse = 3 : 5.

By Pythagoras on the 3–4–5 triple, adjacent = 4.

So, cos(A) = adjacent / hypotenuse = 4 / 5.

Hence, cos(A) = 4/5.

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About This Quiz
Apply Sohcahtoa: Solve For Unknown Sides & Angles - Quiz

Ready to apply the SOHCAHTOA ratios to find missing sides, angles, and distances? This quiz emphasizes practical problem-solving using trigonometric functions with both numeric and algebraic inputs. You will practice converting between ratios and angles, using inverse trig functions, and checking solutions with the Pythagorean identity. Problems include angles of... see moreelevation, slope relationships, and complementary angles in right triangles, reinforcing both conceptual understanding and procedural accuracy.
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2) In a right triangle, one acute angle measures 60°, and the side adjacent to it is 7 units. Find the hypotenuse.

Explanation

cos 60° = adjacent / hypotenuse = 7 / H.

So, H = 7 / cos 60° = 7 / (1/2) = 14.

Hence, the hypotenuse is 7 / cos 60°.

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3) A right triangle has legs 9 and 12. What is the hypotenuse?

Explanation

c² = 9² + 12² = 81 + 144 = 225.

So, c = √225 = 15.

Hence, the hypotenuse is 15.

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4) In △XYZ, right at Z, tan(X) = 5/12. What is sin(X)?

Explanation

Let opposite = 5 and adjacent = 12.

Hypotenuse = √(5² + 12²) = √(25 + 144) = √169 = 13.

So, sin(X) = opposite / hypotenuse = 5 / 13.

Hence, sin(X) = 5/13.

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5) A right triangle has an opposite side of 1.2 m and an adjacent side of 4.0 m. Find the angle θ to the nearest degree.

Explanation

tan θ = 1.2 / 4.0 = 0.3.

θ = arctan(0.3) ≈ 16.7°.

Hence, θ ≈ 17°.

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6) A right triangle has hypotenuse 20 and one leg 12. What is the length of the other leg?

Explanation

Other leg² = 20² − 12² = 400 − 144 = 256.

Other leg = √256 = 16.

Hence, the other leg is 16.

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7) In a right triangle, the hypotenuse is 50 m and one acute angle measures 37°. Find the side opposite that angle.

Explanation

sin 37° = opposite / hypotenuse.

Opposite = 50 · sin 37°.

Hence, the opposite side is 50 sin 37°.

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8) In a right triangle, cos(θ) = 0.8. What is tan(θ) if θ is acute?

Explanation

sin²θ + cos²θ = 1 ⇒ sin θ = √(1 − 0.8²) = √(1 − 0.64) = √0.36 = 0.6.

tan θ = sin θ / cos θ = 0.6 / 0.8 = 0.75.

Hence, tan θ = 0.75.

Submit

9) A 30°–60°–90° triangle has hypotenuse 14. What is the length of the shorter leg?

Explanation

Shorter leg (opposite 30°) = (1/2) × hypotenuse.

Shorter leg = 14 / 2 = 7.

Hence, the shorter leg is 7.

Submit

10) In △ABC (right at C), AC = 5 and BC = 12. What is sin(A)?

Explanation

AB = √(5² + 12²) = √169 = 13.

sin(A) = opposite / hypotenuse = BC / AB = 12 / 13.

Hence, sin(A) = 12/13 

Submit

11) A right triangle has base 6 and height 8, with θ at the base. What is sin(θ)?

Explanation

Hypotenuse = √(6² + 8²) = √100 = 10.

sin θ = opposite / hypotenuse = 8 / 10 = 4 / 5.

Hence, sin θ = 4/5.

Submit

12) Using the same triangle as quesiton 11, what is tan(θ)?

Explanation

tan θ = opposite / adjacent = 8 / 6 = 4 / 3.

Hence, tan θ = 4/3.

Submit

13) A angle θ satisfies tan(θ) = 2. The horizontal distance is 45 m. What is the vertical rise?

Explanation

tan θ = rise / run = rise / 45 = 2.

rise = 2 × 45 = 90 m.

Hence, the vertical rise is 90 m.

Submit

14) A right triangle has angle θ where sin(θ) = 7/25. What is cos(θ)?

Explanation

cos θ = √(1 − sin²θ)

= √(1 − (7/25)²)

= √(1 − 49/625)

= √(576/625) = 24/25.

Hence, cos θ = 24/25.

Submit

15) In a right triangle, the side opposite angle A is 35 m, and angle A = 12°. Find the adjacent side to the nearest meter.

Explanation

tan A = opposite / adjacent ⇒ adjacent = opposite / tan A.

adjacent = 35 / tan 12°.

Hence, adjacent = 35 / tan 12°.

Submit

16) In a right triangle, the longer leg is 9 and the hypotenuse is 15. What is the measure of the angle opposite the longer leg to the nearest degree?

Explanation

sin θ = opposite / hypotenuse = 9 / 15 = 3 / 5.

θ = arcsin(3/5) ≈ 36.9°.

Hence, θ ≈ 37°.

Submit

17) In a right triangle, the legs have a ratio of rise : run = 1 : 12. If the rise is 0.75 m, find the run.

Explanation

run = 12 × rise = 12 × 0.75 = 9 m.

Hence, the run is 9 m.

Submit

18) A right triangle has sides in the ratio 5:12:13. Which statement is true?

Explanation

With legs 5 and 12, hypotenuse 13:

sin(opposite 12) = 12/13 is valid.

Hence, sin(θ) can be 12/13.

Submit

19) A right triangle has an opposite side of 9 ft and a hypotenuse of 15 ft. Find the measure of the angle opposite the 9 ft side to the nearest degree.

Explanation

sin θ = 9 / 15 = 3 / 5.

θ = arcsin(3/5) ≈ 36.9°.

Hence, θ ≈ 37°.

Submit

20) A triangle has an angle θ with adjacent side 10 and opposite side 7. Which expression gives θ?

Explanation

tan θ = opposite / adjacent = 7 / 10.

θ = arctan(7/10).

Hence, θ = arctan(7/10).

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 right triangle ABC with right angle at C, if sin(A) = 3/5, what is...
In a right triangle, one acute angle measures 60°, and the side...
A right triangle has legs 9 and 12. What is the hypotenuse?
In △XYZ, right at Z, tan(X) = 5/12. What is sin(X)?
A right triangle has an opposite side of 1.2 m and an adjacent side of...
A right triangle has hypotenuse 20 and one leg 12. What is the length...
In a right triangle, the hypotenuse is 50 m and one acute angle...
In a right triangle, cos(θ) = 0.8. What is tan(θ) if θ is acute?
A 30°–60°–90° triangle has hypotenuse 14. What is the length of...
In △ABC (right at C), AC = 5 and BC = 12. What is sin(A)?
A right triangle has base 6 and height 8, with θ at the base. What is...
Using the same triangle as quesiton 11, what is tan(θ)?
A angle θ satisfies tan(θ) = 2. The horizontal distance is 45 m....
A right triangle has angle θ where sin(θ) = 7/25. What is cos(θ)?
In a right triangle, the side opposite angle A is 35 m, and angle A =...
In a right triangle, the longer leg is 9 and the hypotenuse is 15....
In a right triangle, the legs have a ratio of rise : run = 1 : 12. If...
A right triangle has sides in the ratio 5:12:13. Which statement is...
A right triangle has an opposite side of 9 ft and a hypotenuse of 15...
A triangle has an angle θ with adjacent side 10 and opposite side 7....
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