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The tangent at a point P to a given curve is defined as the limiting position of the secant PQ as the point Q approaches P along the curve; whether the point Q is taken on one side of the point P. In trigonometry, the tangent for an angle is denoted by the ratio of cos and sine function; that is an acute angle of a right-handed triangle and it is not equal to the zero. The definition of the Sine operator describes the length of the inverse side divided by the breadth of the hypotenuse side of the right-angle triangle.
Tangents
This is Sinθ= (inverse side of the right-handed triangle) (hypotenuse side of the right-handed triangle) and the cos operator denoted by the Cosθ= (Adjacent side of the right-handed triangle) (hypotenuse side of the right-handed triangle). Therefore, the tangent function is dependent on the Cos and Sin operator; Tanθ describes the slope between the altitude and hypotenuse of the right-handed triangle. Tangent comes from the wolfram languages and its related function is called hyperbolic tangent denoted as, tanhz= ( ez- e-z ) ( ez+ e-z).
Consider, an equation of the curve be y=f(x) and also consider the given point A on the curve be (x, y) and other neighboring point B on that curve be (x+△x, y+△y). Therefore, the equation of the curve AB is (Y-y) =(y+△y-y) (x+ △x) (X-x) that is △y/△x (X-x). Therefore, the final equation of the tangent is Y-y = dy/dx (X-x) is not parallel to the y-axis and the value of dy/dx is finite.
Different types of tangents
There are mainly six types of tangents available in trigonometry. First was fixed tangent; the definition of the fixed tangent is, it does not change its present control points however it no longer changes as key frames around their moving surfaces. After the tangent was manually moved then the fixed tangents are arising in the trigonometry. If the present control points are set on the horizontal side of the key frame then the intraframe is mostly used as a flat tangent. The third kind of tangent is linear tangent and it takes place when the control points are significantly in line with the intraframe. It acts as an anchor point for the line segment and total control points are linear then the line segment acts as a straight line from the first intra frame to the second intra frame.
Then the first and middlemost keyframe play as a linear tangent. Control point made those lines that remain parallel to the line formed by the intraframe. When the control points are placed vertically with the lines formed by the keyframe and the direction of control points is vertically below the keyframe. The last type of tangent is plateau tangents used in smooth and flat tangents relative to their location of the keyframe.
Properties of Tangents
Tangents have different properties; it is discussed below in this section. The first property of the tangent is that the maximum tangent acts as a curve at a particular point and it shows the approximate slope of the curve at that particular point. The second property of the tangent is, it barely intersects the curve at only one point. At that point there the tangent lies on the curve, which is always differentiable at that particular point.
Consider, OP and OQ are two different radii of a circle and AP and AQ are two tangents to the circle at the points P and Q respectively. Then the proof of angle POQ and angle PAQ are supplementary angles which means the angle POQ + angle PAQ = 180 °. For proofing the above property of the tangent then consider first, the angle OPA =90 degrees and the angle OQA also 90- degrees because the tangent is always perpendicular to the radius of the circle. The addition of the angles OPA, PAQ, OQA and POQ is 360- degrees. The values of the angles OPA and OQA is 90-degree, therefore, the angle POQ and angle PAQ = 180- degree.
Process the tangents to the circle
Construct a tangent to the circle by drawing with a particular radius with the origin. Then join the origin of the circle, after this considers a point on this circle and join it. It gives the diameter of the circle and finally constructs a line that is perpendicular towards the radius of the circle. The answer of the length of the tangent in the circle provides the process that the tangent to the circle.
Let two line segments of length 12 centimeters and length 18 centimeters. Therefore, the total length is (12+18) centimeter = 30 centimeter. According to the properties of the tangent, the length of the tangent = first length+ second length of the line segment in the circle that is (12 18) centimeters.
Key theorems of tangents
Tangent is always perpendicular towards the radius at a particular point on the circle. The equation Y-y = dy/dx (X-x) of the tangent can be written as the form of Y=dy/dx X+(y- x dy/dx), that being of the form of “y=mx+c”. Where, dy/dx = the tangent of the curve. Another important theorem in tangent is, if two tangents of the same lengths are drawn from the outside of any circle then the diameter of that circle is parallel towards the radius of the circle.
Conclusion
The word Tangent is more important in geometry and also in trigonometry. Tangents intersect a given solid or curve at the same point and related geometrical objects are known as a tangent plane or tangent line respectively. If the line segment is not differentiable at that point, then no identified slope occurs on that point.
