Solve Equations with Inverse Tangent

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

Explanation

The question asks for the angle whose tangent is 1.

Since tan(π/4) = 1 and π/4 lies within the range (−π/2, π/2),

arctan(1) = π/4.

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About This Quiz
Solve Equations With Inverse Tangent - 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.

For a negative value, reflect across the x-axis → tan(−π/3) = −√3.

Since −π/3 is within the 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 restricted so it returns one unique angle.

That range is (−π/2, π/2), because tangent is undefined at ±π/2.

Hence, the correct range is (−π/2, π/2).

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

Explanation

The arctan range is (−π/2, π/2), and the endpoints are excluded.

Thus, 0, −π/3, and π/6 are valid, but π/2 is not in the range.

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5) Evaluate arctan(√3) in degrees.

Explanation

tan(60°) = √3.

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

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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 angle is within the range (−90°, 90°).

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7) Evaluate tan(arctan(−2)).

Explanation

tan and arctan are inverse functions.

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

Hence, tan(arctan(−2)) = −2.

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

Explanation

Simplify the equation: 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 the angle whose tangent equals 0 is 0.

Therefore, arctan(0) = 0.

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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.

Hence, θ = arctan(1).

This equals 45° or π/4 radians.

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11) Evaluate arctan(−1) in radians.

Explanation

tan(−π/4) = −1.

Since −π/4 is within (−π/2, π/2), arctan(−1) = −π/4.

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12) Solve tan(x) = 0 for the principal value.

Explanation

tan(x) = 0 when x = 0, π, 2π, etc.

The principal value (in −π/2 to π/2) is 0.

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13) Solve for x (principal value): tan(x) = 5.

Explanation

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

Thus, x = arctan(5).

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

Explanation

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

Rounded to two decimals, it’s 0.93 radians.

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

Explanation

tan and arctan cancel each other out for all real numbers.

Thus, tan(arctan(x)) = x always holds true.

Other options have restrictions or are not always valid.

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

Explanation

Simplify: 2tan(θ) = 2√3 → tan(θ) = √3.

The angle whose tangent equals √3 is π/3.

Hence, θ = π/3.

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

Explanation

tan(π/6) = √3/3.

Since the value is negative, the angle is −π/6 in the range (−π/2, π/2).

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

tan(θ) = rise/run = 18/36 = 1/2.

So θ = arctan(18/36).

This gives the correct angle of elevation.

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20) Evaluate arctan(√3/3) in degrees.

Explanation

tan(30°) = √3/3.

Thus, arctan(√3/3) = 30°.

It 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.
Solve tan(x) = 0 for the principal value.
Solve for x (principal value): 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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