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Trigonometric Ratio (Tangent) Calculator

Calculate the tangent of an angle in a right-angled triangle.

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Tangent of Theta

Formula first

Overview

The tangent ratio represents the relationship between the lengths of the two legs of a right-angled triangle. It is defined as the ratio of the length of the side opposite a given acute angle to the length of the side adjacent to that same angle.

Symbols

Variables

Opp = Opposite Side, Adj = Adjacent Side, () = Tangent of Theta

Opp
Opposite Side
Variable
Adj
Adjacent Side
Variable
Tangent of Theta
Variable

Apply it well

When To Use

When to use: This equation is used in right-angled trigonometry when the hypotenuse is either unknown or not required for the calculation. It is particularly effective for determining heights of objects or distances along the ground when an angle of elevation or depression is provided.

Why it matters: The tangent function is the mathematical foundation for understanding slopes and gradients in coordinate geometry. In the real world, it is indispensable for navigation, surveying land, and engineering structures like ramps and staircases.

Avoid these traps

Common Mistakes

  • Confusing opposite and adjacent.
  • Convert units and scales before substituting, especially percentages, time units, or powers of ten.
  • Interpret the answer with its unit and context; a percentage, rate, ratio, and physical quantity do not mean the same thing.

One free problem

Practice Problem

Practice Problem 1

A surveyor measures the distance from a point on the ground to the base of a tower as 50 meters. If the tangent of the angle of elevation to the top of the tower is 1.2, what is the height of the tower?

Adjacent Side50
Tangent of Theta1.2

Solve for: opp

Hint: Rearrange the formula to solve for the opposite side by multiplying the tangent value by the adjacent side.

Practice Problem 2

A ramp is designed with a vertical rise of 3 meters and a horizontal run of 12 meters. Calculate the tangent of the angle the ramp makes with the ground.

Opposite Side3
Adjacent Side12

Solve for:

Hint: The tangent is simply the ratio of the vertical rise (opposite) to the horizontal run (adjacent).

Practice Problem 3

The shadow of a tree is measured. If the tree is 15 meters tall and the tangent of the sun's angle of elevation is 0.75, how long is the shadow cast on the ground?

Opposite Side15
Tangent of Theta0.75

Solve for: adj

Hint: To find the adjacent side, divide the opposite side length by the tangent value.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Wikipedia: Trigonometric functions
  2. Britannica: Tangent (trigonometry)
  3. Britannica, The Editors of Encyclopaedia. 'Tangent'. Encyclopedia Britannica, 26 Oct. 2023.
  4. Wikipedia, 'Trigonometric functions'
  5. Britannica: Trigonometric function
  6. AQA GCSE Maths — Trigonometry