50+ trigonometric ratios sin cos tan csc sec and cot worksheets ... - Free Printable

Let’s solve each question one by one with clear explanations.

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Question 1: Which of the following trigonometric functions does NOT use the hypotenuse in its ratio?

Options:
- A. secant
- B. tangent
- C. sine
- D. cosine

✔ Correct Answer: B. tangent

Explanation:
In a right triangle, for an acute angle θ:

- sine (sin θ) = opposite / hypotenuse → uses hypotenuse
- cosine (cos θ) = adjacent / hypotenuse → uses hypotenuse
- secant (sec θ) = 1/cos θ = hypotenuse / adjacent → uses hypotenuse
- tangent (tan θ) = opposite / adjacent → does NOT use hypotenuse

So, tangent is the only one that doesn’t involve the hypotenuse.

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Question 2: What trigonometric ratio corresponds to the ratio of the side adjacent to acute angle in a right triangle to the side opposite to it?

Options:
- A. Sine
- B. Cosine
- C. Tangent
- D. Cotangent

✔ Correct Answer: D. Cotangent

Explanation:
- tan θ = opposite / adjacent
- So, cot θ = adjacent / opposite — which is exactly what the question describes.

Cotangent is the reciprocal of tangent.

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Question 3: Which of these trigonometric ratios is the reciprocal of sec θ?

Options:
- A. sin θ
- B. cos θ
- C. tan θ
- D. csc θ

✔ Correct Answer: B. cos θ

Explanation:
By definition:
> sec θ = 1 / cos θ

Therefore, the reciprocal of sec θ is cos θ.

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Question 4: In ΔECQ, ∠Q is a right angle. Which side of the triangle is the hypotenuse?

Options:
- A. EC
- B. QC
- C. EQ
- D. None of the above

✔ Correct Answer: A. EC

Explanation:
In any right triangle, the hypotenuse is the side opposite the right angle.

Since ∠Q is the right angle, the side opposite to it is EC (connecting points E and C).

So, EC is the hypotenuse.

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Question 5: Given triangle ZNE with right angle at N, sides labeled:
- ZN = 5 (vertical leg)
- NE = 12 (horizontal leg)
- ZE = 13 (hypotenuse)

Which statement is CORRECT?

Options:
- A. sin Z = 12/13
- B. tan Z = 5/12
- C. cos E = 13/12
- D. sec E = 12/13

✔ Correct Answer: A. sin Z = 12/13

Explanation:

Let’s analyze each option carefully.

First, label the triangle properly:
- Right angle at N
- So, vertices: Z, N, E — with right angle at N.
- Side ZN = 5 (adjacent to angle Z, opposite to angle E)
- Side NE = 12 (opposite to angle Z, adjacent to angle E)
- Hypotenuse ZE = 13

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Option A: sin Z = 12/13

- For angle Z:
- Opposite side = NE = 12
- Hypotenuse = ZE = 13
- So, sin Z = opposite/hypotenuse = 12/13 ✔ CORRECT

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Option B: tan Z = 5/12

- tan Z = opposite / adjacent = NE / ZN = 12 / 5 → not 5/12 ✘

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Option C: cos E = 13/12

- cos E = adjacent / hypotenuse
- For angle E:
- Adjacent side = NE = 12
- Hypotenuse = 13
- So, cos E = 12/13 — not 13/12 ✘ (also, cosine can’t be >1)

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Option D: sec E = 12/13

- sec E = 1 / cos E = 13/12 — not 12/13 ✘

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✔ So, only Option A is correct.

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## ✔ Final Answers:

1. B. tangent
2. D. Cotangent
3. B. cos θ
4. A. EC
5. A. sin Z = 12/13

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