What is the tangent of 45°?

The tangent of an angle is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. For a 45° angle in a right triangle, the opposite and adjacent sides are of equal length.

Since the tangent function is given by the formula:

[

\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

]

for a 45° angle, this results in:

[

\tan(45°) = \frac{\text{length of opposite side}}{\text{length of adjacent side}} = \frac{x}{x} = 1

]

where (x) is any positive length. Therefore, the tangent of 45° is equal to 1.

This principle also aligns with the unit circle where at 45°, the coordinates of the point are ((\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})). The tangent can also be calculated as the y-coordinate divided by the x-coordinate, which again equals:

[

\tan(45°) = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2