tangent

Examples of tangent in the following topics:

• Sine, Cosine, and Tangent

• Tangent Planes and Linear Approximations

  • The tangent line (or simply the tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.
  • When the curve is given by $y = f(x)$ the slope of the tangent is $\frac{dy}{dx}$, so by the point–slope formula the equation of the tangent line at $(x_0, y_0)$ is:
  • The tangent plane to a surface at a given point $p$ is defined in an analogous way to the tangent line in the case of curves.
  • Note the similarity of the equations for tangent line and tangent plane.
  • Since a tangent plane is the best approximation of the surface near the point where the two meet, tangent plane can be used to approximate the surface near the point.

• Tangent as a Function

  • Characteristics of the tangent function can be observed in its graph.
  • The tangent function can be graphed by plotting $\left(x,f(x)\right)$ points.
  • The shape of the function can be created by finding the values of the tangent at special angles.
  • At these values, the graph of the tangent has vertical asymptotes.
  • As with the sine and cosine functions, tangent is a periodic function.

• Tangent Vectors and Normal Vectors

  • A vector is normal to another vector if the intersection of the two form a 90-degree angle at the tangent point.
  • In order for a vector to be normal to an object or vector, it must be perpendicular with the directional vector of the tangent point.
  • Tangent vectors are almost exactly like normal vectors, except they are tangent instead of normal to the other vector or object.
  • A plane can be determined as normal to the object if the directional vector of the plane makes a right angle with the object at its tangent point.
  • This plane is normal to the point on the sphere to which it is tangent.

• The Derivative and Tangent Line Problem

  • The tangent line $t$ (or simply the tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.
  • If $k$ is known, the equation of the tangent line can be found in the point-slope form:
  • Using derivatives, the equation of the tangent line can be stated as follows:
  • The line shows the tangent to the curve at the point represented by the dot.
  • Define a derivative as the slope of the tangent line to a point on a curve

• Optimization of consumption

  • An agent should consume goods at the point where the most preferred available indifference curve is tangent to their budget constraint.
  • In an economy with only two products An individual consumer should choose to consume goods at the point where the most preferred available indifference curve is tangent to their budget constraint .
  • That is, the indifference curve tangent to the budget constraint represents the maximum utility obtained utilizing the entire budget of the consumer.
  • The tangent point represents the amount of goods the consumer should purchase to fully utilize their budget to obtain maximum utility.
  • Output is optimized where the budget constraint, marked in blue, is tangent to indifference curve, marked in red.

• Inverse Trigonometric Functions

  • However, the sine, cosine, and tangent functions are not one-to-one functions.
  • The graph of the tangent function is limited to $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.
  • The inverse tangent function $y = \tan^{-1}x$ means $x = \tan y$.
  • The inverse tangent function can also be written $\arctan x$.
  • The arctangent function is a reflection of the tangent function about the line $y = x$.

• Trigonometric Functions

  • The most familiar trigonometric functions are the sine, cosine, and tangent .
  • In the context of the standard unit circle with radius 1, where a triangle is formed by a ray originating at the origin and making some angle with the x-axis, the sine of the angle gives the length of the y-component (rise) of the triangle, the cosine gives the length of the x-component (run), and the tangent function gives the slope (y-component divided by the x-component) .
  • The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side: so called because it can be represented as a line segment tangent to the circle, that is the line that touches the circle, from Latin linea tangens or touching line (cf. tangere, to touch).
  • The sine, tangent, and secant functions of an angle constructed geometrically in terms of a unit circle.

• Sine, Cosine, and Tangent

  • A common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of “Sine is opposite over hypotenuse (Soh), Cosine is adjacent over hypotenuse (Cah), Tangent is opposite over adjacent (Toa).”
  • Remembering the mnemonic, "SohCahToa", the sides given are opposite and adjacent or "o" and "a", which would use "T", meaning the tangent trigonometric function.
  • Use the acronym SohCahToa to define Sine, Cosine, and Tangent in terms of right triangles

• Tangent and Velocity Problems

  • Slope of tangent of position or displacement time graph is instantaneous velocity and its slope of chord is average velocity.
  • Recognize that the slope of a tangent line to a curve gives the instantaneous velocity at that point in time