tangent

(noun)

A straight line touching a curve at a single point without crossing it.

Related Terms

Examples of tangent in the following topics:

  • Sine, Cosine, and Tangent

  • 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.

  • 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 $y = x$.

  • 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

  • Special Angles

    • Note that while only sine and cosine are defined directly by the unit circle, tangent can be defined as a quotient involving these two:
    • Tangent functions also have simple expressions for each of the special angles.
    • Let's find the tangent of $60^{\circ}$.
    • In quadrant III, “Trig,” only tangent is positive.
    • Since tangent functions are derived from sine and cosine, the tangent can be calculated for any of the special angles by first finding the values for sine or cosine.

  • Trigonometric Symmetry Identities

    • On the other hand, sine and tangent are odd functions because they are symmetric about the origin.
    • The sine, cosecant, tangent, and cotangent functions are symmetric about the origin.
    • Now that we know the sine, cosine, and tangent of $\displaystyle{\frac{5\pi}{6}}$, we can apply the symmetry identities to find the functions of $\displaystyle{-\frac{5\pi}{6}}$.
    • Sine, cosecant, tangent, and cotangent are odd functions, and are symmetric around the origin.

  • Secant and the Trigonometric Cofunctions

    • We have discussed three trigonometric functions: sine, cosine, and tangent.
    • The cotangent function is the reciprocal of the tangent function, and is abbreviated as $\cot$.
    • Recall that we used values for the sine and cosine functions to calculate the tangent function.
    • Define the trigonometric functions that are the reciprocals of sine, cosine, and tangent

  • Graphs of Exponential Functions, Base e

    • The graph of $e^x$ has the property that the slope of the tangent line to the graph at each point is equal to its $y$-coordinate at that point.

  • Circles as Conic Sections

    • Tangent: a straight line that touches the circle at a single point.

  • Defining Trigonometric Functions on the Unit Circle

    • Applying this formula, we can find the tangent of any angle identified by a unit circle as well.
    • For the angle $t$ identified in the diagram  of the unit circle showing the point $\displaystyle{ \left(-\frac{\sqrt2}{2}, \frac{\sqrt2}{2}\right) }$, the tangent is: