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How can tangent be used to find the height or distance of an object?

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Diagram of a right triangle labeled with sides a, b, and hypotenuse c, alongside formulas for the six trigonometric functions-sine, cosine, tangent, cosecant, secant, and cotangent-each defined by ratios of the triangle's sides. — trigonometry Based on the definitions, various simple relationships exist among the functions. For example, csc A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A, and tan A = sin A/cos A.

In trigonometry, the tangent function is a powerful tool for calculating the height or distance of an object by modeling the scenario as a right triangle.12

For an angle $A$ within a right triangle, the tangent is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle:2

$\tan A = \frac{length of side opposite angle A}{length of side adjacent to angle A}$

To find an unknown dimension, you can rearrange this formula:

To find height (opposite side): If you know the distance from the object (adjacent side) and the angle of elevation, you can calculate the height using:
$Height = Distance \times \tan (Angle)$
To find distance (adjacent side): If you know the height of the object and the angle of elevation, you can calculate the distance using:
$Distance = \frac{Height}{\tan (Angle)}$

This method relies on the fact that the tangent ratio depends solely on the size of the angle, allowing for accurate measurements without needing to physically reach the top of an object or traverse the entire distance to it.1