Quotient Derivation Quiz: Deriving Quotients from Unit Circle and Right Triangles

  • 10th Grade
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| Questions: 20 | Updated: Dec 16, 2025
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1) Simplify: (cos^2θ)/(sinθ·cosθ).

Explanation

(cos^2θ)/(sinθ·cosθ) = [cosθ·cosθ]/[sinθ·cosθ] = cosθ/sinθ = cotθ, provided sinθ and cosθ are nonzero.

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About This Quiz
Quotient Derivation Quiz: Deriving Quotients From Unit Circle And Right Triangles - Quiz

How do quotient identities take shape from basic geometric ideas? In this quiz, you’ll derive tangent and cotangent step by step using unit-circle definitions and triangle-based ratios. You’ll compare side lengths, interpret coordinate movements, and connect each symbolic relationship back to a visual understanding. Through guided reasoning, you’ll see why... see morethese quotients work the way they do. By the end, you’ll feel more confident using identities because you genuinely understand their origins.
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2) Tan(−θ) = −tanθ because y/x changes sign when y changes to −y while x stays the same.

Explanation

At (x, y) = (cosθ, sinθ), replacing θ by −θ gives (x, −y). Then tan(−θ) = (−y)/x = −(y/x) = −tanθ.

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3) Select all expressions that equal 1 whenever defined.

Explanation

Each of A, B, C, and D simplifies to 1 by reciprocal or cancellation when denominators are nonzero. cotθ − tanθ is generally not 1.

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4) A unit circle point is (√3/2, 1/2). Find tanθ.

Explanation

tanθ = y/x = (1/2)/(√3/2) = 1/√3.

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5) Select all expressions equal to tanθ.

Explanation

Triangle: tanθ = opposite/adjacent. Unit circle: tanθ = y/x since y=sinθ and x=cosθ. Also secθ/cscθ = (1/cosθ)/(1/sinθ) = sinθ/cosθ = tanθ. adjacent/opposite is cotθ.

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6) A unit circle point is (√2/2, −√2/2). Compute tanθ.

Explanation

Here x = √2/2 and y = −√2/2. Then tanθ = y/x = (−√2/2)/(√2/2) = −1.

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7) In a right triangle with opposite=5 and adjacent=12 for angle θ, what is tanθ?

Explanation

By definition tanθ = opposite/adjacent. Here tanθ = 5/12.

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8) A unit circle point is (−√2/2, √2/2). Find cotθ.

Explanation

cotθ = x/y = (−√2/2)/(√2/2) = −1.

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9) If tanθ = 7/24 for an acute θ, then cotθ = 24/7.

Explanation

cotθ is the reciprocal of tanθ: cotθ = 1/tanθ = 24/7.

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10) On the unit circle, tanθ equals y/x where (x, y) = (cosθ, sinθ).

Explanation

On the unit circle x = cosθ and y = sinθ. Therefore tanθ = sinθ/cosθ = y/x, for x ≠ 0.

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11) In the same triangle (opposite=5, adjacent=12), what is cotθ?

Explanation

cotθ = adjacent/opposite = 12/5.

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12) Given sinθ = 3/5 and cosθ = 4/5, write tanθ.

Explanation

Use tanθ = sinθ/cosθ = (3/5)/(4/5) = 3/4.

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13) Given sinθ = 3/5 and cosθ = 4/5, write cotθ.

Explanation

cotθ = cosθ/sinθ = (4/5)/(3/5) = 4/3.

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14) In a 8-15-17 right triangle (opposite=8, adjacent=15, hypotenuse=17 for angle θ), which are true?

Explanation

tanθ = opposite/adjacent = 8/15; cotθ = adjacent/opposite = 15/8; therefore tanθ·cotθ = (8/15)(15/8) = 1.

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15) If sinθ = 5/13 and cosθ = 12/13, evaluate tanθ.

Explanation

tanθ = sinθ/cosθ = (5/13)/(12/13) = 5/12.

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16) If tanθ = opposite/adjacent, then cotθ = opposite/adjacent as well.

Explanation

cotθ is the reciprocal ratio: cotθ = adjacent/opposite, not opposite/adjacent.

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17) Given tanθ = 3/4, which option equals y/x for the corresponding unit circle point?

Explanation

On the unit circle, tanθ = y/x. If tanθ = 3/4, then y/x = 3/4.

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18) Select all expressions equal to cotθ.

Explanation

cotθ = adjacent/opposite (triangle) = cosθ/sinθ = x/y on the unit circle, and also 1/tanθ. opposite/adjacent is tanθ.

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19) Tanθ is defined whenever sinθ is defined.

Explanation

tanθ = sinθ/cosθ requires cosθ ≠ 0. Even when sinθ exists, tanθ is undefined at cosθ = 0 (e.g., θ = π/2 + kπ).

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20) Simplify: (sin^2θ)/(sinθ·cosθ).

Explanation

(sin^2θ)/(sinθ·cosθ) = [sinθ·sinθ]/[sinθ·cosθ] = sinθ/cosθ = tanθ, provided sinθ and cosθ are nonzero.

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Simplify: (cos^2θ)/(sinθ·cosθ).
Tan(−θ) = −tanθ because y/x changes sign when y changes to −y...
Select all expressions that equal 1 whenever defined.
A unit circle point is (√3/2, 1/2). Find tanθ.
Select all expressions equal to tanθ.
A unit circle point is (√2/2, −√2/2). Compute tanθ.
In a right triangle with opposite=5 and adjacent=12 for angle θ,...
A unit circle point is (−√2/2, √2/2). Find cotθ.
If tanθ = 7/24 for an acute θ, then cotθ = 24/7.
On the unit circle, tanθ equals y/x where (x, y) = (cosθ, sinθ).
In the same triangle (opposite=5, adjacent=12), what is cotθ?
Given sinθ = 3/5 and cosθ = 4/5, write tanθ.
Given sinθ = 3/5 and cosθ = 4/5, write cotθ.
In a 8-15-17 right triangle (opposite=8, adjacent=15, hypotenuse=17...
If sinθ = 5/13 and cosθ = 12/13, evaluate tanθ.
If tanθ = opposite/adjacent, then cotθ = opposite/adjacent as well.
Given tanθ = 3/4, which option equals y/x for the corresponding unit...
Select all expressions equal to cotθ.
Tanθ is defined whenever sinθ is defined.
Simplify: (sin^2θ)/(sinθ·cosθ).
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