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Tangent lines are a geometric relationship between a line and a curve, determining the slope of a curve at a point. Circles have an infinite number of tangents, and spheres have tangent planes. Trigonometry defines the tangent of an angle in a triangle. Tangents are used in calculus to observe the instantaneous rate of change at a given moment.
A tangent line is a geometric relationship between a line and a curve such that the curve and line share only one common point. The tangent line is always on the outside or convex side of the curve. It is impossible to draw a tangent inside a curve or circle. Tangents determine the slope of a curve at a point. They play a role in geometry, trigonometry and calculus.
Every circle has an infinite number of tangents. The four tangents of a circle that are 90 degrees apart comprise a square that inscribes the circle. In other words, a circle can be drawn inside an exact square and will touch the square at four points. Knowing this is useful for solving many geometry problems involving areas.
Spheres can also have a tangent line, although it is more common to speak of a tangent plane sharing only one common point with the sphere. An infinite number of tangent lines could pass through that point of intersection, all of which would be contained in the tangent plane. These concepts are used in troubleshooting volume problems. A sphere can be placed inside a cube. If the diameter of the cube equals the side length of the cube, remembering that all sides are equal in a cube, the sphere will share six common points with the cube.
In trigonometry, the tangent of an angle of a triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. The triangle is formed by the radii of two radii from the center of a circle. The first ray forms the base of the triangle and the second ray extends to intersect the tangent line of the first. Grade is often defined as rise over ride. Thus, the tangent, or slope, of the line connecting the two rays is the same as in the trigonometric identity.
When considering a line tangent to a curve, unless the curve is the arc of a circle, an observer should note the point of intersection. This is because the curve is not of constant radius. An example of this would be the flight path of a baseball after being hit by a bat.
The ball will accelerate away from the club, but will reach its peak and descend under gravity. The flight path will have the shape of a parabola. The tangent to the curve at any point will produce the velocity of the ball at that moment.
This mathematical description of the slope of a curve of inconstant curvature is fundamental to the study of calculus. The calculation allows you to observe the instantaneous rate of change at a given moment. This is useful for checking process reaction rates, rocket fuel consumption for spacecraft launches, or exactly where to be to catch a baseball.
