Input (tan θ)	Arctan (Degrees)	Arctan (Radians)
0	0°	0 rad
0.57735 (√3/3)	30°	π/6 rad (≈ 0.5236)
1	45°	π/4 rad (≈ 0.7854)
1.73205 (√3)	60°	π/3 rad (≈ 1.0472)
Infinity (approach from +)	90°	π/2 rad (≈ 1.5708)
-0.57735 (-√3/3)	-30° or 330°	-π/6 rad (≈ -0.5236) or 11π/6 rad
-1	-45° or 315°	-π/4 rad (≈ -0.7854) or 7π/4 rad

Understanding How to Find Arctan on Scientific Calculator

This article provides a deep dive into the arctangent function, often represented as arctan, atan, or tan⁻¹, and explains how to find arctan on scientific calculator, both manually and using our online tool.

What is Arctan (Inverse Tangent)?

The arctangent, or inverse tangent, is the inverse function of the tangent function in trigonometry. If you have a value ‘x’ that is the tangent of an angle ‘θ’ (i.e., tan(θ) = x), then the arctangent of ‘x’ will give you the angle ‘θ’ (i.e., arctan(x) = θ). It essentially answers the question: “What angle has a tangent equal to this value?”

For example, we know that tan(45°) = 1. Therefore, arctan(1) = 45° (or π/4 radians). It’s crucial to note that the arctangent function typically returns a principal value within a specific range, usually -90° to +90° (-π/2 to +π/2 radians), to ensure it’s a true function (one input gives one output).

Who Should Use Arctan?

Students: Learning trigonometry and solving problems involving angles and ratios.
Engineers: Calculating angles in structures, electronics, and various fields.
Physicists: Analyzing vectors, forces, and wave phenomena.
Programmers: In graphics, robotics, and simulations involving rotations and directions.
Surveyors: Determining angles of elevation or depression and land slopes.

Common Misconceptions

A common mistake is thinking that tan⁻¹(x) is the same as 1/tan(x) (which is cot(x)). The “-1” in tan⁻¹ signifies the inverse function, not a reciprocal.

Arctan Formula and Mathematical Explanation

The fundamental relationship is:

If tan(θ) = x, then arctan(x) = θ

Where ‘x’ is the value of the tangent (often representing a ratio like y/x in coordinates or slope), and ‘θ’ is the angle whose tangent is ‘x’. The result ‘θ’ is usually given in radians or degrees. The principal value of arctan(x) is always in the range (-π/2, π/2) radians or (-90°, 90°).

When dealing with coordinates (x, y) to find the angle, you often use `arctan(y/x)`. However, to get the correct angle in all four quadrants (0 to 360° or 0 to 2π radians), the `atan2(y, x)` function is preferred in programming and many calculators, as it considers the signs of both y and x.

Variables Table

Variable	Meaning	Unit	Typical Range
x (or input value)	The value for which arctan is calculated (the tangent of an angle)	Dimensionless	-∞ to +∞
θ (or result)	The angle whose tangent is x	Degrees or Radians	-90° to 90° or -π/2 to π/2 (principal value)

Practical Examples (Real-World Use Cases)

Example 1: Finding the Angle of a Ramp

Suppose a ramp has a rise (vertical height) of 2 meters and a run (horizontal distance) of 5 meters. The slope (or tangent of the angle of inclination θ) is rise/run = 2/5 = 0.4.

To find the angle θ, we calculate arctan(0.4).

Using our calculator with input 0.4 and unit degrees: arctan(0.4) ≈ 21.8°.

So, the ramp makes an angle of about 21.8 degrees with the horizontal.

Example 2: Angle from Vector Components

A force vector has components Fx = 30 N and Fy = 40 N. The angle θ this vector makes with the positive x-axis can be found using arctan(Fy/Fx) = arctan(40/30) = arctan(1.333…).

Using the calculator with input 1.3333: arctan(1.3333) ≈ 53.13°.

The angle is approximately 53.13 degrees. (Note: `atan2(40, 30)` would directly give this result in the first quadrant).

How to Use This Arctan Calculator

Enter Value: Input the number for which you want to find the arctangent into the “Enter Value” field. This number represents the tangent of the angle you are looking for.
Select Unit: Choose whether you want the resulting angle to be displayed in “Degrees” or “Radians” from the dropdown menu.
Calculate: The calculator updates the result in real time as you type or change the unit. You can also click the “Calculate” button.
Read Results: The primary result (the angle) is shown prominently. You also see the input value and the unit you selected.
Reset: Click “Reset” to clear the input and set the unit back to the default (Degrees).
Copy Results: Click “Copy Results” to copy the input, unit, and calculated angle to your clipboard.

Understanding how to find arctan on scientific calculator is simple with this tool. Just input the tangent value, and the angle is calculated.

Key Factors That Affect Arctan Results

Input Value: The value you enter directly determines the angle. Larger positive values approach 90° (π/2 radians), while large negative values approach -90° (-π/2 radians).
Unit Selection (Degrees/Radians): The same input will yield different numerical results depending on whether you choose degrees or radians. 1 radian ≈ 57.3 degrees.
Calculator Precision: The number of decimal places the calculator (or our tool) uses can slightly affect the result’s precision.
Principal Value Range: Standard arctan functions return values between -90° and +90°. If you need angles outside this range (e.g., 0° to 360°), you need to consider the signs of the components (like in atan2(y,x)).
Domain of Arctan: The arctan function is defined for all real numbers as input.
Range of Arctan: The output (principal value) is strictly between -90° and +90° (-π/2 and +π/2 radians), exclusive of the endpoints for the standard function when considering infinity.

Frequently Asked Questions (FAQ)

1. How do I find arctan on a physical scientific calculator?: Most scientific calculators have a “tan” button. The arctan function is usually its secondary function, accessed by pressing a “Shift”, “2ndF”, or “Inv” key first, then the “tan” button (it’s often labeled as tan⁻¹ above the tan button). Enter the value, then press Shift/2ndF + tan.
2. What is the difference between arctan(x) and tan⁻¹(x)?: There is no difference; they are just different notations for the same inverse tangent function.
3. What is the range of the arctan function?: The principal value range of arctan(x) is (-90°, 90°) or (-π/2, π/2 radians).
4. Can I find the arctan of a negative number?: Yes. For example, arctan(-1) = -45° or -π/4 radians.
5. What is the arctan of infinity?: As the input value approaches positive infinity, arctan approaches 90° (π/2 radians). As it approaches negative infinity, arctan approaches -90° (-π/2 radians).
6. Is arctan the same as 1/tan?: No. 1/tan(x) is cot(x) (cotangent), which is the reciprocal of the tangent. arctan(x) is the inverse function of tangent.
7. Why do I need to choose between degrees and radians?: Angles can be measured in degrees or radians. The numerical value of the angle will be different depending on the unit used (e.g., 45° = π/4 radians ≈ 0.7854 radians).
8. What is atan2(y, x) and how is it related to arctan?: atan2(y, x) is a two-argument function that computes the arctangent of y/x but uses the signs of both y and x to determine the correct quadrant for the resulting angle, typically returning a value between -180° and 180° (-π and π radians). It’s more robust for converting Cartesian coordinates (x,y) to polar coordinates (r, θ).

Related Tools and Internal Resources

Explore other calculators that might be helpful:

Sine Calculator: Calculate the sine of an angle.
Cosine Calculator: Calculate the cosine of an angle.
Tangent Calculator: Calculate the tangent of an angle.
Right Triangle Calculator: Solve for sides and angles of a right triangle.
Degrees to Radians Converter: Convert angles between degrees and radians.
Slope Calculator: Calculate the slope between two points, related to the tangent of the angle of inclination.

Understanding how to find arctan on scientific calculator is a fundamental skill in trigonometry, and our arctan calculator makes it easy.

How To Find Arctan On Scientific Calculator

Arctan Calculator

Arctan (Inverse Tangent) Calculator

Results:

Common Arctan Values