Inverse Tangent: Evaluate & Interpret (Principal Values)

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| Questions: 20 | Updated: Nov 10, 2025
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1) Evaluate arctan(1) in radians.

Explanation

arctan(1) asks for the angle whose tangent is 1.

On the unit circle, tan(π/4) = 1.

Because π/4 lies in the principal range (−π/2, π/2), arctan(1) = π/4.

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About This Quiz
Inverse Tangent: Evaluate & Interpret (Principal Values) - Quiz

Explore how the arctangent (arctan) function links slopes, ratios, and angles. This quiz focuses on evaluating arctan values, identifying the correct principal range (–π/2 to π/2), and interpreting what each result represents on the unit circle. Students practice determining when tangent is positive or negative, finding corresponding angles in radians... see moreor degrees, and applying tan(arctan x) = x to verify results. Ideal for building a strong conceptual understanding of inverse tangent relationships. see less

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2) Solve for x in the principal range: tan(x) = −√3.

Explanation

We know tan(π/3) = √3.

Since tangent is an odd function (tan(−θ) = −tan(θ)), tan(−π/3) = −√3.

Thus, in the principal range (−π/2, π/2), x = −π/3.

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3) What is the range of y = arctan(x)?

Explanation

The arctangent function is defined so that it always gives a unique angle.

To make it a function, its outputs are restricted to (−π/2, π/2).

The endpoints are excluded because tan(±π/2) is undefined.

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4) Which value is NOT in the range of y = arctan(x)?

Explanation

arctan(x) only gives angles between −π/2 and π/2 (not including those endpoints).

0, −π/3, and π/6 are within that range, but π/2 is not.

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5) Evaluate arctan(√3) in degrees.

Explanation

tan(60°) = √3.

Because 60° lies in the arctan range (−90°, 90°), arctan(√3) = 60°.

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6) If tan(θ) = 0.75 and θ = arctan(0.75), what is θ to the nearest tenth of a degree?

Explanation

Use a calculator: arctan(0.75) ≈ 36.87°.

Rounded to the nearest tenth, θ = 36.9°.

This value is within the valid range (−90°, 90°).

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7) Evaluate tan(arctan(−2)).

Explanation

The tangent and arctangent functions are inverses.

So tan(arctan(x)) = x for all real x.

Thus, tan(arctan(−2)) = −2.

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8) Solve 3tan(x) − √3 = 0 for the principal value of x.

Explanation

Simplify: 3tan(x) − √3 = 0 → tan(x) = √3/3 = 1/√3.

The angle whose tangent is 1/√3 is π/6.

Thus, x = π/6.

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9) Evaluate arctan(0) in radians.

Explanation

tan(0) = 0, so arctan(0) = 0.

This is the center of the principal range (−π/2, π/2).

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10) A right triangle has opposite side 7 and adjacent side 7 relative to angle θ. Express θ using inverse tangent.

Explanation

tan(θ) = opposite/adjacent = 7/7 = 1.

Therefore, θ = arctan(1), which equals π/4 radians or 45°.

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11) Evaluate arctan(−1) in radians.

Explanation

tan(−π/4) = −1.

Since −π/4 lies within the arctan range (−π/2, π/2), arctan(−1) = −π/4.

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12) Find the principal value solving tan(x) = 0.

Explanation

tan(x) = 0 when x = 0, π, 2π, etc.

The principal value (the one in −π/2 to π/2) is x = 0.

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13) Find the principal value of x given tan(x) = 5.

Explanation

Arctangent gives the unique angle between −π/2 and π/2 whose tangent is 5.

Therefore, x = arctan(5).

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14) Evaluate arctan(4/3) to the nearest 0.01 radians.

Explanation

On a calculator, arctan(4/3) ≈ 0.9273 radians.

Rounded to two decimal places, that’s 0.93 radians.

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15) Which identity is always true for all real x?

Explanation

tan and arctan undo each other for any real x.

The identity tan(arctan(x)) = x always holds true.

The others only hold in limited ranges or are incorrect.

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16) Solve for θ in (−π/2, π/2): 2tan(θ) = 2√3.

Explanation

Divide both sides by 2 → tan(θ) = √3.

The angle whose tangent is √3 in the principal range is π/3.

Therefore, θ = π/3.

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17) Evaluate arctan(−√3/3) in radians.

Explanation

tan(π/6) = √3/3.

For a negative value, take the reflection below the x-axis: θ = −π/6.

Thus, arctan(−√3/3) = −π/6.

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18) If arctan(a) = π/4, find a.

Explanation

Apply tangent to both sides: a = tan(π/4).

Since tan(π/4) = 1, a = 1.

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19) A ramp rises 18 inches over a horizontal run of 36 inches. Let θ be the angle of elevation. Which expression gives θ?

Explanation

The tangent of an angle equals rise/run = 18/36 = 1/2.

Therefore, θ = arctan(18/36).

This expression gives the correct relationship.

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20) Evaluate arctan(√3/3) in degrees.

Explanation

We know tan(30°) = √3/3.

So, arctan(√3/3) = 30°.

This angle lies within the principal range (−90°, 90°).

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Evaluate arctan(1) in radians.
Solve for x in the principal range: tan(x) = −√3.
What is the range of y = arctan(x)?
Which value is NOT in the range of y = arctan(x)?
Evaluate arctan(√3) in degrees.
If tan(θ) = 0.75 and θ = arctan(0.75), what is θ to the nearest...
Evaluate tan(arctan(−2)).
Solve 3tan(x) − √3 = 0 for the principal value of x.
Evaluate arctan(0) in radians.
A right triangle has opposite side 7 and adjacent side 7 relative to...
Evaluate arctan(−1) in radians.
Find the principal value solving tan(x) = 0.
Find the principal value of x given tan(x) = 5.
Evaluate arctan(4/3) to the nearest 0.01 radians.
Which identity is always true for all real x?
Solve for θ in (−π/2, π/2): 2tan(θ) = 2√3.
Evaluate arctan(−√3/3) in radians.
If arctan(a) = π/4, find a.
A ramp rises 18 inches over a horizontal run of 36 inches. Let θ be...
Evaluate arctan(√3/3) in degrees.
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