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Arctan 1 – Definition in Degrees and Radians

Jo-ann Caballes

Updated on January 12, 2024

Arctan 1 is a fundamental concept in trigonometry, important for understanding the relationship between an angle and the sides of a right triangle. Trigonometry, a branch of mathematics, deals with angles and triangles. Arctan, short for ‘arc tangent,’ is a function that helps us find the measure of an angle when we know the tangent of that angle. Tangent is a ratio of the opposite side to the adjacent side of a right triangle. Specifically, arctan 1 is the angle whose tangent value is 1.

Arctan 1 in Degrees

In geometry, we measure angles in degrees. A full circle is divided into 360 degrees. Therefore, when we express arctan 1 in degrees, we are finding out how many degrees that particular angle has. It is a straightforward way to understand the size of an angle.

Calculating Arctan 1 in Degrees

To calculate arctan 1 in degrees, we use the formula:

Angle in degrees=arctan(1)*(180/π​)

​Given that arctan⁡(1) is π/4​ radians, the calculation would be (π/4)​×(180º/π)​=45º. Therefore, arctan 1 equals 45º degrees.

Arctan 1 in Radians

Radians are another unit to measure angles, based on the radius of a circle. One radian is the angle at the center of a circle where the arc length is equal to the radius of the circle.

Calculating Arctan 1 in Radians

The calculation of arctan 1 in radians is more direct as the arctan function in trigonometry is inherently in radians. The value of arctan 1 in radians is π/3​ or approximately 0.7854 radians.

Arctan 1 in Terms of Pi

Pi (π) is a mathematical constant representing the ratio of a circle’s circumference to its diameter. Expressing arctan 1 in terms of π helps in visualizing and understanding the concept in the context of circular measurements. As mentioned, arctan 1 equals π/4​.

Arctan Table

An arctan table is a reference tool that lists various tangent values and their corresponding angles in degrees and radians. For example, the table will show that a tangent value of 1 corresponds to an angle of 45 degrees or π/4​ radians.

Practical Examples of Arctan 1

1. Measuring the Slope of a Roof: Suppose an architect needs to design a roof that slopes upwards at a consistent angle. If the roof rises 1 meter for every 1 meter it extends horizontally, the slope forms a right triangle. The tangent of the angle is the ratio of the rise to the run, which is 1/1 = 1. To find the angle of the slope, we use arctan 1. This gives us an angle of 45 degrees or π/4 radians. This is practical in architectural design for ensuring consistency in roof slopes.

2. Calculating the Angle of a Ramp: Imagine a wheelchair ramp where the height of the ramp and the horizontal distance covered are equal. If the ramp rises 2 meters over a horizontal distance of 2 meters, the tangent of the angle of elevation is 2/2 = 1. Using arctan 1, we determine that the ramp makes a 45-degree angle with the ground. This calculation is crucial in construction to make sure ramps meet accessibility standards.

3. Determining the Angle of a Kite String: Consider a child flying a kite where the string is pulled taut, and the kite is directly above a point that is horizontally equal in distance to the length of the string. If the string is 50 meters long and the kite is vertically 50 meters high, the tangent of the angle between the string and the ground is 50/50 = 1. Using arctan 1, we find that the string is at a 45-degree angle to the ground. This example can be used in basic physics or geometry classes to explain angles in a fun and relatable way.

4. Angles in Road Inclines: In road construction, engineers often need to determine the steepness of an incline. If a hill rises 10 meters over a horizontal distance of 10 meters, the tangent of the incline’s angle is 10/10 = 1. Using arctan 1, the angle of the incline is calculated to be 45 degrees or π/4 radians. This example is significant for transportation engineering, ensuring roads are safe and meet specific grade requirements.

Frequently Asked Questions on Arctan 1

What is the significance of arctan 1 in math?

Arctan 1 is significant as it helps in determining specific angles in trigonometric functions, crucial for solving geometrical problems and understanding the relationships in right triangles.

How often is arctan 1 used in real-world applications?

Arctan 1 is frequently used in various fields, including engineering, architecture, and physics, where precise angle measurements are necessary.

Is arctan 1 different in degrees and radians?

The value of arctan 1 is the same conceptually but is expressed differently in degrees and radians. In degrees, it’s 45 degrees, and in radians, it’s π/4.

Can arctan 1 be used in navigation or mapping?

Yes, arctan 1 can be used in navigation and mapping, especially in calculating angles of elevation or depression, which are important for geographical measurements and in aviation.

Why do we use radians in some calculations instead of degrees?

Radians are often used in calculations involving calculus and higher mathematics because they provide a more natural approach to measuring angles in terms of the properties of a circle.

Jo-ann Caballes

6 articles

As a seasoned educator with a Bachelor’s in Secondary Education and over three years of experience, I specialize in making mathematics accessible to students of all backgrounds through Brighterly. My expertise extends beyond teaching; I blog about innovative educational strategies and have a keen interest in child psychology and curriculum development. My approach is shaped by a belief in practical, real-life application of math, making learning both impactful and enjoyable.