The tangent of an angle, denoted as tan x, can be expressed in multiple ways.
Understanding Tangent
The tangent function (tan) is a fundamental trigonometric ratio that relates the sides of a right triangle to one of its acute angles. It's important for many applications in mathematics, physics, and engineering.
Tangent Formulas
The reference provides these key formulas for tan x:
- tan x = (opposite side) / (adjacent side) - This is the definition of the tangent based on the sides of a right-angled triangle.
- tan x = 1 / (cot x) - This relates the tangent to the cotangent function.
Expressing tan x
Using the formulas, we can definitively state the following:
- tan x as a Ratio of Sides:
- In a right triangle, if 'x' represents one of the acute angles, then
tan xis the ratio of the length of the side opposite the anglexto the length of the side adjacent to the anglex.
- In a right triangle, if 'x' represents one of the acute angles, then
- tan x in relation to cot x:
- The tangent of an angle is the reciprocal of the cotangent of the same angle, meaning
tan x = 1 / (cot x).
- The tangent of an angle is the reciprocal of the cotangent of the same angle, meaning
Practical Application
- Example 1: Consider a right triangle with an angle x, where the opposite side has a length of 3 and the adjacent side has a length of 4. Then tan x = 3 / 4 = 0.75
- Example 2: If cot x = 2 then tan x = 1 / 2 = 0.5
Summary
| Expression | Definition |
|---|---|
| tan x = Opposite / Adjacent | Ratio of sides |
| tan x = 1 / cot x | Reciprocal of cotangent |
