Right Triangle - Unknown Angle using Tangent

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Right Triangle - Unknown Angle using Tangent

Right Triangle - Unknown Angle using Tangent
\[\theta=\tan^{-1}\left(\frac{opposite}{adjacent}\right)\]

Variables

theta = unknown angle of the right triangle (rad or °)
opposite = length of the opposite side (m)
adjacent = length of the adjacent side (m)

Description

What is this formula?


This trigonometric formula calculates an unknown angle in a right triangle using the inverse tangent function.


The tangent ratio relates the opposite side of an angle to its adjacent side. Using the inverse tangent function allows the angle to be determined from known side lengths.


When to use it


Use this formula when:


- The opposite side is known

- The adjacent side is known

- An acute angle must be calculated

- Solving right triangle problems

- Working with slopes, elevations, or inclinations


The resulting angle may be expressed in degrees or radians depending on calculator settings.


Example


If:


opposite = 7 m

adjacent = 10 m


Then:


theta = atan(7/10)


theta = atan(0.7)


theta ≈ 34.99°


Applications


- Trigonometry and geometry

- Engineering calculations

- Surveying and navigation

- Physics and mechanics

- Terrain slope analysis

- Vector decomposition

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