Direct Calculation from Triangles

  • 10th Grade
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| Attempts: 16 | Questions: 20 | Updated: Dec 2, 2025
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1) If sin θ = 5/13, then csc θ = ?

Explanation

Cosecant is the reciprocal of sine. Since sin θ is 5/13, csc θ is 13/5.

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About This Quiz
Direct Calculation From Triangles - Quiz

Are you ready to put your triangle skills to the test? In this quiz, you’ll practice moving between sine, cosine, tangent, and their reciprocals—cosecant, secant, and cotangent—by working directly with right triangles. Take this quiz to sharpen your ability to compute reciprocal ratios quickly and confidently.

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2) If cos θ = 8/17, then sec θ = ?

Explanation

Secant is the reciprocal of cosine. Since cos θ is 8/17, sec θ is 17/8.

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3) If csc θ = -25/7, then sin θ = ?

Explanation

Sine is the reciprocal of cosecant. Since csc θ is −25/7, sin θ is −7/25.

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4) If csc θ = 25/7, which of the following cannot be a trigonometric ratio in the same right triangle?

Explanation

From csc θ = 25/7 we get sin θ = 7/25 and cos θ = 24/25 (valid). A value like cos θ = 25/24 is bigger than 1, so it can’t happen.

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5) If csc θ = 17/8, then sin θ = ?

Explanation

Since csc θ is 17/8, sin θ is 8/17.

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6) If sec θ = 25/24, then cos θ = ?

Explanation

Since sec θ is 25/24, cos θ is 24/25.

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7) If sin θ = -12/13, then csc θ = ?

Explanation

Since sin θ is −12/13, csc θ is −13/12.

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8) If cos θ = 3/5, then sec θ = ?

Explanation

Since cos θ is 3/5, sec θ is 5/3.

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9) If csc θ = 2, then sin θ = ?

Explanation

If csc θ is 2, then sin θ is 1/2.

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10) If sec θ = 10, then cos θ = ?

Explanation

If sec θ is 10, then cos θ is 1/10.

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11) In a right triangle, cos θ = 12/13. Find csc θ.

Explanation

With cos θ = 12/13 (a 5–12–13 triangle), sin θ is 5/13, so csc θ is 13/5.

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12) If tan θ = 5/12, which of the following cannot be a trigonometric ratio in the same right triangle?

Explanation

For tan θ = 5/12 (the 5–12–13 triangle), csc θ should be 13/5, not 12/5. So 12/5 is the one that can’t occur.

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13) If cot θ = 5/12, then tan θ = ?

Explanation

Cotangent and tangent are reciprocals. If cot θ is 5/12, then tan θ is 12/5.

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14) In a right triangle, sin θ = 15/17. Find sec θ.

Explanation

With sin θ = 15/17, cos θ is 8/17, so sec θ is 17/8.

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15) A right triangle has legs 7 and 24, and hypotenuse 25. Find cot θ if θ is the angle opposite the side of length 7.

Explanation

Cotangent is adjacent over opposite. With opposite 7 and adjacent 24, cot θ is 24/7.

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16) A right triangle has legs 8 and 15, and hypotenuse 17. Find sec θ if θ is the angle opposite the side of length 8.

Explanation

In the 8–15–17 triangle (θ opposite 8), cos θ is 15/17, so sec θ is 17/15.

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17) If csc θ = 41/40, then sin θ = ?

Explanation

If csc θ is 41/40, then sin θ is 40/41.

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18) If sec θ = 13/5, which of the following cannot be a trigonometric ratio in the same right triangle?

Explanation

If sec θ is 13/5, then cos θ is 5/13 and sin θ must be 12/13 (or negative, by quadrant). So sin θ = 5/13 doesn’t match that triangle.

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19) A right triangle has legs 9 and 12, and hypotenuse 15. Find csc θ if θ is the angle opposite the side of length 9.

Explanation

With sides 9–12–15 and θ opposite 9, sin θ is 3/5, so csc θ is 5/3.

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20) In a right triangle, tan θ = 3/4. Find csc θ.

Explanation

tan θ = 3/4 matches a 3–4–5 triangle, so sin θ is 3/5 and csc θ is 5/3.

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If sin θ = 5/13, then csc θ = ?
If cos θ = 8/17, then sec θ = ?
If csc θ = -25/7, then sin θ = ?
If csc θ = 25/7, which of the following cannot be a...
If csc θ = 17/8, then sin θ = ?
If sec θ = 25/24, then cos θ = ?
If sin θ = -12/13, then csc θ = ?
If cos θ = 3/5, then sec θ = ?
If csc θ = 2, then sin θ = ?
If sec θ = 10, then cos θ = ?
In a right triangle, cos θ = 12/13. Find csc θ.
If tan θ = 5/12, which of the following cannot be a...
If cot θ = 5/12, then tan θ = ?
In a right triangle, sin θ = 15/17. Find sec θ.
A right triangle has legs 7 and 24, and hypotenuse 25. Find cot...
A right triangle has legs 8 and 15, and hypotenuse 17. Find sec...
If csc θ = 41/40, then sin θ = ?
If sec θ = 13/5, which of the following cannot be a...
A right triangle has legs 9 and 12, and hypotenuse 15. Find csc...
In a right triangle, tan θ = 3/4. Find csc θ.
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