How do you find the angles that satisfy tan(x) = 1?

To find the angles that satisfy tan(x) = 1, use x = 45° + 180°n or x = π/4 + πn, where n is an integer.

To understand why these angles work, let's start with the basic properties of the tangent function. The tangent of an angle in a right-angled triangle is the ratio of the opposite side to the adjacent side. When tan(x) = 1, it means the opposite and adjacent sides are equal, which happens at 45° (or π/4 radians) in the first quadrant.

However, the tangent function is periodic with a period of 180° (or π radians). This means that the tangent of an angle repeats its values every 180°. Therefore, if tan(x) = 1 at 45°, it will also be 1 at 45° + 180° = 225°, and so on. This can be generalised to x = 45° + 180°n, where n is any integer (0, ±1, ±2, ...).

In radians, the same principle applies. The angle 45° is equivalent to π/4 radians. So, the angles that satisfy tan(x) = 1 in radians are given by x = π/4 + πn, where n is any integer.

By using these formulas, you can find all the angles that satisfy the equation tan(x) = 1. For example, if n = 0, you get 45° or π/4. If n = 1, you get 225° or 5π/4, and so on. This method ensures you cover all possible solutions within the infinite set of angles where the tangent function equals 1.