Trigonometric functions

In a right triangle, each of the three primary trigonometric functions is a ratio of two sides taken relative to one of the acute angles. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent — the relationships captured by the mnemonic SOHCAHTOA. The reciprocal functions follow directly: cosecant is 1/sin, secant is 1/cos, and cotangent is 1/tan. Because the ratios depend only on the angle and not on the triangle's size, similar triangles give identical values.

The unit circle extends these definitions beyond acute angles. For a point where the terminal side of an angle θ meets a circle of radius 1 centered at the origin, cos θ is the x-coordinate and sin θ is the y-coordinate, and tan θ = sin θ / cos θ is the slope of that terminal side. This lets the functions accept any angle, positive or negative, in degrees or radians, and it explains their signs in each quadrant. It also produces the standard exact values worth memorizing, such as sin 30° = 1/2, cos 60° = 1/2, and sin 45° = cos 45° = √2/2.

Graphed, sine and cosine are smooth waves that repeat every 360° or 2π radians, oscillating between −1 and 1; they are identical curves shifted by 90°. Tangent repeats every 180° or π radians and has vertical asymptotes wherever cosine equals zero. The most-used identity is the Pythagorean identity sin²θ + cos²θ = 1, which follows immediately from the unit circle definition and the Pythagorean theorem.

Trigonometric functions appear on several exams at different depths. The ACT tests basic right-triangle trigonometry and function properties, the AMC 12 uses them in advanced problem solving, and AP Calculus AB requires their derivatives along with the special limits — such as the limit of sin x / x as x approaches 0 — that those derivatives depend on.