Tangent Definitions and Examples

Tangent Definitions, Formulas, & Examples

GET TUTORING NEAR ME!

(800) 434-2582

By submitting the following form, you agree to Club Z!'s Terms of Use and Privacy Policy

    Home / Educational Resources / Math Resources / Tangent Definitions and Examples

    Tangent Definitions and Examples

    You’re driving along, minding your own business, when suddenly you see a sign for a place called Tangent. What does it mean? A tangent is defined as “a straight line or plane that touches a curve or surface at only one point,” but that doesn’t really help you if you’re still trying to figure out what it is. In this blog post, we will explore tangents in greater depth with examples to help you understand the concept.

    What is a tangent?

    A tangent is a straight line or plane that touches a curve or surface at only one point, without crossing it or intersecting it.

    Tangent Meaning

    In mathematics, a tangent is a line that touches a curve but doesn’t cross it. For example, the line shown in red in the diagram below is tangent to the circle at the point where it meets the circle.

    A tangent can also be a straight line that just grazes a surface without actually touching it. The edge of a table is tangent to the floor, for instance.

    The word “tangent” comes from the Latin word for “touch.”

    Tangent of a Circle

    A tangent of a circle is a line that touches the circle at only one point. A secant line is a line that intersects the circle at two points. The length of the tangent line can be found using the following formula:

    Tangent = Circle Radius x Secant Length

    The following are examples of tangents and secants:

    As you can see in the image, the red line is the tangent and the blue line is the secant. The tangent line is shorter than the secant line.

    Point of Tangency

    In mathematics, the point of tangency is the point where a line touches a curve. The line is said to be tangent to the curve at that point. The word “tangent” comes from the Latin word for “touch.”

    Tangent lines are important in calculus because they can be used to find the rate of change of a function at a particular point. For example, if you were trying to find how fast water was flowing out of a container at a certain time, you could draw a tangent line to the curve representing the water level in the container over time. The steepness of the tangent line would tell you how fast the water was flowing.

    There are two ways to calculate the slope of a tangent line: using limits or derivatives. Limits are usually easier to calculate, but derivatives give you more information about what is happening at the point of tangency.

    To find the equation of a tangent line, you need to know the coordinates of the point of tangency and the slope of the line. Then you can use any of various methods for finding equations of lines, such as points-slope form or slope-intercept form.

    Tangent Properties

    A tangent is a straight line that touches a curve at a single point. The word comes from the Latin word for “touch.”

    The slope of a tangent line is the same as the slope of the curve at the point of tangency. This is because the tangent line is just a straight line approximation of the curve at that point.

    You can find the equation of a tangent line using calculus. First, find the derivative of the equation of the curve. Then, plug in the coordinates of the point of tangency to find the slope of the tangent line. Finally, use this slope and the coordinates of the point of tangency to write an equation for the Tangent line.

    Tangent Theorems

    In mathematics, a tangent is a line that touches a curve at a single point. The word “tangent” comes from the Latin word for “touch”.

    There are two main types of tangents: internal and external. Internal tangents touch the curve at only one point, while external tangents touch the curve at two points.

    The most common type of tangent is the straight line tangent. A straight line tangent is simply a straight line that touches the curve at only one point.

    There are also other types of tangents, such as circle tangents and parabola tangents. Circle tangents touch the curve at only one point, but they are not straight lines. Parabola tangents touch the curve at two points and are also not straight lines.

    Tangent theorems are mathematical results that deal with properties of tangents to curves. The most famous Tangent Theorem is probably the Pythagorean Theorem for right triangles, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

    Tangent of Circle Formula

    A circle is a two-dimensional shape with a center point and radius. The radius is the distance from the center to the edge of the circle. The circumference is the distance around the edge of the circle. The area of a circle is the space inside the circle.

    The tangent of a circle is a line that touches the edge of the circle at only one point. A tangent line is perpendicular to the radius at the point where it touches the circumference. The formula for finding the tangent of a circle is:

    tangent = opposite / adjacent

    Where opposite is the length of the line segment from the center of the circle to the point where it intersects with the tangent line, and adjacent is the length of line segment from that same point to either side of thecircle.

    Types of tangents

    There are three types of tangents that can be used in geometry:

    1. Secant Tangent: A secant tangent is a line that intersects a curve at two points.

    2. External Tangent: An external tangent is a line that intersects a curve at only one point, and this point is not on the curve itself.

    3. Internal Tangent: An internal tangent is a line that intersects a curve at only one point, and this point is on the curve itself.

    Examples of tangents

    There are many types of tangents, but here are a few examples to help you understand what a tangent is:

    A straight line tangent to a curve at a point is the straight line that just touches the curve at that point.

    A circle is said to be tangent to another circle, or to a line, if it meets the other circle or line only at one point.

    Two lines in a plane are called parallel if they do not meet even if they are extended indefinitely in both directions. A line that intersects one of the parallel lines at some point other than their point of intersection is called a transversal. If the transversal intersects both lines at right angles, then the two lines are said to be orthogonal.

    When to use a tangent

    A tangent is a straight line that touches a curve at only one point. In other words, it is the line that “just barely” touches the graph of a function at some point.

    There are many situations in which you might want to use a tangent. For instance, if you are finding the slope of a graph at some point, you will need to use a tangent. Other times, you might want to find the equation of a tangent line in order to better understand the behavior of a function near some point.

    Tangents can be used in calculus to help find limits and derivatives. They can also be used in physics to describe the motion of objects along curved paths. In any situation where you are dealing with a curve, it is often helpful to consider its tangent lines.

    How to use a tangent

    There are a few different ways to use a tangent. The most common way is to find the equation of a tangent line. To do this, you will need to find the slope of the tangent line. This can be done by taking the derivative of the function at the point where the tangent line intersects the curve. Once you have the slope, you can use any point on the tangent line to write the equation of the line using point-slope form.

    Another way to use a tangent is to approximate values for a function that is not easy to directly evaluate. This is often done using calculus and is called numerical integration. To do this, you will take small segments of the curve and approximate them with straight lines. The sum of all of these approximations will give you an estimate for the value of the original function.

    You can also use a tangent to create a graphical representation of a function. This is often done in cases where it is not possible or difficult to find an analytic solution (a mathematical expression for the function). In this case, you can take points on the curve and connect them with straight lines to create a Tangent Function Graph.

    Tips for using tangents

    There are a few things to keep in mind when using tangents in your writing. First, be sure that the topic you’re tangenting off of is something that you’re knowledgeable about. Second, make sure the tangent is relevant to the main point of your article. Third, keep your tangents brief and to the point. Fourth, don’t forget to come back to the main point of your article after you’ve made your tangent. Fifth, proofread your work to ensure that your tangents make sense and flow well with the rest of your writing. following these tips will help you use tangents effectively and improve the overall quality of your writing.

    Conclusion

    A tangent is a line or curve that touches a given curve at only one point and does not intersect it. The word tangent comes from the Latin word for “touch.” Tangents are important in mathematics because they can be used to define the slope of a curve at any given point. The slope of a tangent is also known as the derivative of the function at that point. In calculus, derivatives are used to find rates of change, which can be applied to problems in physics and engineering.

    Tangent

    Plots

    Plots

    Plots

    Alternate form assuming x is real

    sin(2 x)/(cos(2 x) + 1)

    Alternate forms

    sin(x)/cos(x)

    (i (e^(-i x) - e^(i x)))/(e^(-i x) + e^(i x))

    Roots

    x = π n, n element Z

    Integer root

    x = 0

    Properties as a real function

    {x element R : x/π + 1/2 not element Z}

    R (all real numbers)

    periodic in x with period π

    surjective onto R

    odd

    Series expansion at x = 0

    x + x^3/3 + (2 x^5)/15 + O(x^6) (Taylor series)

    Derivative

    d/dx(tan(x)) = sec^2(x)

    Indefinite integral

    integral tan(x) dx = -log(cos(x)) + constant (assuming a complex-valued logarithm)

    Identities

    tan(x) = tan(m π + x) for m element Z

    tan(x) = -cot(2 x) + csc(2 x)

    tan(x) = (1 - cos(2 x)) csc(2 x)

    tan(x) = sin(2 x)/(1 + cos(2 x))

    tan(x) = -(2 tan(x/2))/(-1 + tan^2(x/2))

    tan(x) = (2 tan(x/2))/(1 - tan^2(x/2))

    tan(x) = (x sqrt(-tan^2(x)))/sqrt(-x^2)

    tan(x) = (sec(x/3) sin(x))/(-1 + 2 cos((2 x)/3))

    Alternative representations

    tan(x) = 1/cot(x)

    tan(x) = cot(π/2 - x)

    tan(x) = -cot(π/2 + x)

    Series representations

    tan(x) = i + 2 i sum_(k=1)^∞ (-1)^k q^(2 k) for q = e^(i x)

    tan(x) = i sum_(k=-∞)^∞ (-1)^k e^(2 i k x) sgn(k)

    tan(x) = -i + 2 i sum_(k=0)^∞ (-1)^k e^(-2 i (1 + k) x) for Im(x)<0

    Integral representations

    tan(x) = integral_0^x sec^2(t) dt

    tan(x) = 2/π integral_0^∞ (-1 + t^((2 x)/π))/(-1 + t^2) dt for 0<Re(x)<π/2

    Find the right fit or it’s free.

    We guarantee you’ll find the right tutor, or we’ll cover the first hour of your lesson.