In trigonometry, the law of tangents is a rule describing the relationship between the tangents of two angles of a triangle, and opposite-side lengths of the triangle. In the 1200s, Persian mathematician Nasir al-Din al-Tusi developed the law of tangents for spherical triangles. The tangent of an angle is the ratio of the length of the adjacent side to the length of the opposite of a triangle.
A tangent line touches a curve at one single point. (A tangent line is assumed to be a straight, one dimensional object, while curvature refers to any deviation from a truly straight, or flat figure.) The slope-intercept formula for a line is y = mx + b, where m is the slope of the line, and b is the y-intercept. The point-slope formula addresses both the point on the line where the line and curve touch, as well as the slope of the line.
Tangent planes are two-dimensional figures, having length and width. For the purposes of curved tangents, think of an orange with a small square of glass perched on top. The single point where the two touch is the tangent of the curvature of the orange and the plane of glass. The equation for this tangent plane is fx(x0y0)(x'x0)+fy(x0y0)(y'y0)'(z'z0)=0.
Trigonometric functions, including tangents, have a variety of vital, real-world applications. Such mathematical functions aid in navigation, engineering, and physics. The "surface normal" (addressing curved surfaces and the tangent plane) is used in computer graphics for purposes of shading. Tangents help to determine angles of elevation and depression, which is important information when considering building a structure adjacent to a hillside or slope.