Unit Circle Interpretation Quiz: Fundamental Shape and Unit Circle Interpretation

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| Questions: 20 | Updated: Dec 16, 2025
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1) Between any two consecutive vertical asymptotes, the graph of y = tanθ is strictly increasing.

Explanation

On each interval (−π/2 + kπ, π/2 + kπ), tanθ has derivative sec^2θ > 0, so it increases strictly across the interval.

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About This Quiz
Unit Circle Interpretation Quiz: Fundamental Shape And Unit Circle Interpretation - Quiz

Curious about what the unit circle truly represents? In this quiz, you’ll explore how the circle’s geometry connects angles, coordinates, and trigonometric values. You’ll interpret key points, examine symmetry, and learn how radius-one geometry reveals sine and cosine directly. Through visual reasoning and guided examples, you’ll build a deeper understanding... see moreof why the unit circle is so central to trigonometry, helping you read angles and evaluate values with confidence and accuracy.
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2) Select all angles in [−π/2, π/2) where tanθ > 1.

Explanation

On [−π/2, π/2), tanθ increases from −∞ to +∞ and crosses 1 at θ = π/4. Thus tanθ > 1 for θ in (π/4, π/2): examples include π/3 and 3π/8. The others give tanθ ≤ 1.

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3) Write the general formula for the vertical asymptotes of y = tanθ.

Explanation

tanθ is undefined when cosθ = 0, which happens at θ = π/2 + kπ, k ∈ ℤ.

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4) The graph of y = tanθ passes through the origin.

Explanation

tan0 = 0, so the point (0, 0) lies on y = tanθ.

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5) Select all expressions equivalent to tanθ (where defined).

Explanation

On the unit circle, tanθ = y/x = sinθ/cosθ; also secθ/cscθ = (1/cosθ)/(1/sinθ) = sinθ/cosθ = tanθ. The expressions cosθ/sinθ = cotθ and −sinθ/cosθ = −tanθ.

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6) For θ with cosθ ≠ 0, the slope of the line from the origin to (cosθ, sinθ) equals

Explanation

Slope = rise/run = y/x = sinθ/cosθ = tanθ when cosθ ≠ 0.

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7) Which statement about the tangent graph is true?

Explanation

tanθ = 0 when sinθ = 0 and cosθ ≠ 0, i.e., θ = kπ. It is odd and has period π, not 2π.

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8) On [0, 2π), in which intervals is tanθ > 0?

Explanation

Since tanθ = sinθ/cosθ, tanθ is positive when sinθ and cosθ have the same sign: Quadrants I and III, i.e., (0, π/2) and (π, 3π/2).

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9) If (x, y) = (cosθ, sinθ) = (√3/2, 1/2), what is tanθ?

Explanation

tanθ = y/x = (1/2)/(√3/2) = 1/√3.

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10) Evaluate tan(45°).

Explanation

On the unit circle, sin45° = √2/2 and cos45° = √2/2, so tan45° = (√2/2)/(√2/2) = 1.

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11) Tan(−θ) = −tanθ, so y = tanθ is an odd function symmetric about the origin.

Explanation

sin is odd and cos is even, so tan(−θ) = sin(−θ)/cos(−θ) = (−sinθ)/(cosθ) = −tanθ. This is the definition of an odd function.

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12) Evaluate tan(135°).

Explanation

At 135°, sin135° = √2/2 and cos135° = −√2/2, so tan135° = (√2/2)/(−√2/2) = −1.

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13) Which angle is a zero of y = tanθ on [0, 2π)?

Explanation

Zeros occur when sinθ = 0 and cosθ ≠ 0. On [0, 2π), θ = 0, π are zeros. Among the options, 0 is a zero.

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14) The range of y = tanθ is all real numbers.

Explanation

As θ approaches the vertical asymptotes, tanθ tends to ±∞, and by continuity on each interval it takes every real value exactly once per period.

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15) Solve tanθ = √3 on [0, 2π). Select all solutions.

Explanation

tanθ = √3 at reference angle π/3. Tangent is positive in Quadrants I and III, giving θ = π/3 and 4π/3 on [0, 2π).

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16) State the sign of tanθ in Quadrant III.

Explanation

In Quadrant III, both sinθ and cosθ are negative, so tanθ = sinθ/cosθ is positive.

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17) What is the fundamental period of y = tanθ?

Explanation

Using tanθ = sinθ/cosθ and angle addition, tan(θ+π) = tanθ for all θ where defined. Hence the smallest positive period is π.

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18) Select all θ in [0, 2π) where tanθ is undefined.

Explanation

tanθ = sinθ/cosθ is undefined when cosθ = 0. On [0, 2π) that occurs at θ = π/2 and 3π/2.

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19) One full basic cycle of y = tanθ is most naturally graphed on which interval?

Explanation

Between consecutive asymptotes at −π/2 and π/2, the graph runs through one complete increasing branch crossing the origin. The function repeats every π.

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20) Tanθ = sinθ/cosθ is defined exactly when cosθ ≠ 0.

Explanation

By definition tanθ = sinθ/cosθ. Division by zero is not allowed, so tanθ is defined precisely at angles with cosθ ≠ 0.

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Between any two consecutive vertical asymptotes, the graph of y =...
Select all angles in [−π/2, π/2) where tanθ > 1.
Write the general formula for the vertical asymptotes of y = tanθ.
The graph of y = tanθ passes through the origin.
Select all expressions equivalent to tanθ (where defined).
For θ with cosθ ≠ 0, the slope of the line from the origin to...
Which statement about the tangent graph is true?
On [0, 2π), in which intervals is tanθ > 0?
If (x, y) = (cosθ, sinθ) = (√3/2, 1/2), what is tanθ?
Evaluate tan(45°).
Tan(−θ) = −tanθ, so y = tanθ is an odd function symmetric about...
Evaluate tan(135°).
Which angle is a zero of y = tanθ on [0, 2π)?
The range of y = tanθ is all real numbers.
Solve tanθ = √3 on [0, 2π). Select all solutions.
State the sign of tanθ in Quadrant III.
What is the fundamental period of y = tanθ?
Select all θ in [0, 2π) where tanθ is undefined.
One full basic cycle of y = tanθ is most naturally graphed on which...
Tanθ = sinθ/cosθ is defined exactly when cosθ ≠ 0.
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