Mathematics
Tangent of a Circle
The tangent of a circle is a line that intersects the circle at exactly one point, known as the point of tangency. This line is perpendicular to the radius of the circle at the point of tangency. In trigonometry, the tangent function represents the ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle.
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6 Key excerpts on "Tangent of a Circle"
eBook - PDFMath Dictionary With Solutions
A Math Review
- Chris Kornegay(Author)
- 1999(Publication Date)
- SAGE Publications, Inc(Publisher)
472 I Tt >· Table, in trigonometry See Trigonometry Table, how to use. >· Tangent, in trigonometry The tangent is one of the six functions of trigonometry. De-fined for a right triangle, it is a fraction of the opposite side to the adjacent side. From angle A, tan A = | . From angle B, tanS = ^. For more, see Trigonometry, right triangle definition of. >• Tangent Line In relation to a curve, a tangent line is any line that intersects the curve in one point without passing through the curve. It touches the curve at one point. In geometry, if we consider a circle, a tangent line is any line that contains a point of the circle but no interior points of circle. It touches the circle at one point. In the figure, we say that (read the line PQ) is tangent to the circle Ο at the point P. ^ Term A term is a number, a variable, or the product of a number and a variable(s). The concept of a term is usually used to delineate one term from another. In the expression 6x^ — — Axy + 5, there are four terms: dx^, —x^, -4xy, and 5. Each term is separated by, yet includes, a positive (-I-) and a negative (—) sign. The term 6x^ is composed of the number -|-6, called the numerical coefficient or coefficient, and a variable with an exponent of 3. The term —x^ is composed of the coefficient — 1 and a variable with an exponent of 2. The — 1 in this term is usually not written since (—l)^^ is the same as —x^. The term —Axy is composed of the number —4 and the variables χ and y. Finally, the last term 5, which has no variable, is referred to as a constant term. For more, see Coefficient or Combining Like Terms. ^ Terminal Side In trigonometry, the terminal side of an angle is a ray that is determined by rotating the initial side of the angle clockwise or counterclockwise. See Angle, in trigonometry. >• Terminating Decimals See Decimal Numbers, terminating. THIRTY-SIXTY-NINETY DEGREE TRIANGLE • 473 I > Terms See Term.
No longer available |Learn more- Daniel C. Alexander, Geralyn M. Koeberlein, , , Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
ANSWER A B D C CD AB Find the point of intersection of the perpendicular bisectors of the two chords. Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. EXAMPLE 1 GIVEN: In Figure 6.40, has a radius of length 5 at B and FIND: CD SOLUTION Draw radius . By the Pythagorean Theorem, According to Theorem 6.3.1, we know that ; then it follows that . CIRCLES THAT ARE TANGENT In this section, we assume that two circles are coplanar. Although concentric circles do not intersect, they do share a common center. For the concentric circles shown in Figure 6.41, the tangent of the smaller circle is a chord of the larger circle. If two circles touch at one point, they are tangent circles. In Figure 6.42(a), circles P and Q are internally tangent; in Figure 6.42(b), circles O and R are externally tangent. CD 2 4 8 CD 2 BC BC 4 ( BC ) 2 16 25 9 ( BC ) 2 5 2 3 2 ( BC ) 2 ( OC ) 2 ( OB ) 2 ( BC ) 2 OC OB 3 OE CD O 290 CHAPTER 6 ■ CIRCLES Unless otherwise noted, all content on this page is © Cengage Learning. O C D B E Q (a) P (b) O R Figure 6.40 Figure 6.41 Figure 6.43 Figure 6.42 EXS. 1–4 As the definition suggests, the line segment joining the centers of two circles is also commonly called the line of centers of the two circles. In Figure 6.43, or is the line of centers for circles A and B . COMMON TANGENT LINES TO CIRCLES A line, line segment, or ray that is tangent to each of two circles is a common tangent for these circles. If the common tangent does not intersect the line segment joining the centers, it is a common external tangent.
eBook - PDF- Doug French(Author)
- 2004(Publication Date)
- Continuum(Publisher)
Moreover, they have a wide variety of interesting properties and applications, including the link with circles and the wave form of their graphs, that can be pursued at an elementary level. However, simple practical problems like finding the height of a tree using the angle of elevation and finding the angle of a slope from its gradient provide good introductory examples for tangent. Familiarity with a definition of tangent and fluency with procedures for calculating lengths and angles can be developed with the one idea, before the potentially confusing task of choosing between two or three functions arises. The early stages of learning about tangents have much in common with ways of introducing sines and cosines, with similar issues concerning definition and equation solving, so these are not rehearsed in detail again here, except to note that the link with gradient, discussed below, does make tangent as ratio at least as important as tangent as length. None the less, lengths are a simpler starting point. Figure 10.5 shows distances stepped off along a tangent to a circle of unit radius by line segments drawn from the centre of the circle with the angles at the centre in intervals of 10°. The length along the tangent from the point of contact is the tangent of the corresponding angle. The values of these lengths can be measured and then verified using the tangent function on a calculator. Alternatively, in the style of the earlier sines and cosines exercise, the calcu-lator values can be used to plot the points on the tangent so that students observe the angle property for themselves. The triangle on the right in Figure 10.5 is the right-angled triangle with an angle of 60° extracted from the circle diagram. Enlargements of this triangle and others with different angles enable lengths to be calculated in appropriate situations and, when the two shorter sides are given, the angle determined.
eBook - PDFCalculus
Early Transcendentals
- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
79 2 One of the crowning achievements of calculus is its ability to capture continuous motion mathematically, allowing that motion to be analyzed instant by instant. THE DERIVATIVE Many real-world phenomena involve changing quantities—the speed of a rocket, the inflation of currency, the number of bacteria in a culture, the shock intensity of an earthquake, the voltage of an electrical signal, and so forth. In this chapter we will develop the concept of a “derivative,” which is the mathematical tool for studying the rate at which one quantity changes relative to another. The study of rates of change is closely related to the geometric concept of a tangent line to a curve, so we will also be discussing the general definition of a tangent line and methods for finding its slope and equation. 2.1 TANGENT LINES AND RATES OF CHANGE In this section we will discuss three ideas: tangent lines to curves, the velocity of an object moving along a line, and the rate at which one variable changes relative to another. Our goal is to show how these seemingly unrelated ideas are, in actuality, closely linked. TANGENT LINES In Example 1 of Section 1.1 we used an informal argument to find the equation of a tangent line to a curve. However, at that stage in the text we did not have a precise definition of a tangent line. Now that limits have been defined precisely we can give a mathematical definition of the tangent line to a curve y = f (x) at a point P(x 0 , f (x 0 )) on the curve. As illustrated in Figure 2.1.1, the slope m PQ of the secant line through P and a second point Q(x, f (x)) on the graph of f is m PQ = f (x) − f (x 0 ) x − x 0 If we let x approach x 0 , then the point Q will move along the curve and approach the point P. Suppose the slope m PQ of the secant line through P and Q approaches a limit as x → x 0 . In that case we can take the value of the limit to be the slope m tan of the tangent line at P. Thus, we make the following definition. Figure 2.1.1
eBook - PDF- L S Blake(Author)
- 2013(Publication Date)
- Butterworth-Heinemann(Publisher)
when c 2 = a 2 m 2 + b 2 . Substituting we have for the equation of a tangent at any point P: v = mx + J(a 2 m 2 + b 2 ). Mathematics 1/19 The equation to the tangent may also be written in the form: (1.42) -c o s 0 + £ s i n 0 = l . a b The coordinates of the point of contact are (a cos 0, b sin 0), 0 being known as the eccentric angle (see Figure 1.37). 1.5.8.12 Tangents Let the straight line y = mx + c meet the hyperbola x 2 a 2 -y 2 /b 2 = 1; then x 2 /a 2 - (mx + c) 2 /b 2 -1 = 0 will give the points of intersection. The condition for tangency is that the roots of this equation are equal, i.e. c = v (a 2 m 2 -b 2 ) and the equation of the tangent is given by y = mx + yj(a 2 m 2 -b 2 ) at any point. Alternati-vely, the tangent to the hyperbola at (JC, y x ) is given by xxja 2 -yyjb 2 = 1. 1.5.8.7 Normal Substituting the value of m above in the general equation for the normal PN to a curve at point P (JC, y x ) given by: (y-y ] )m + (x-x ] ) = 0 we have as the equation for the normal (y-y,)b 2 jy l = (x-x i )a 2 /x r 1.5.8.8 General properties (1) The circle AV 2 BV, is termed the auxiliary circle (Figure 1.37). (2) O M X O T = A 2 . (3) F 2 N = eF 2 P. (4) F,N = *>F,P. (5) PN bisects ^F,PF 2 . (6) The perpendiculars from F,, F 2 to any tangent meet the tangent on the auxiliary circle. 1.5.8.13 Normal The equation for the normal at any point (JC, >,) on the curve is given by: (v -})b 2 l + (x - x ] )a 2 /x ] = 0. (1.44) 1.5.8.14 Asymptotes The tangent to the hyperbola becomes an asymptote when the roots of the equation x 2 /a 2 - (mx + c) 2 /b 2 -1 = 0 are both infinite, i.e. when b 2 -a 2 m 2 = 0 and a 2 mc = 0. Therefore: rn= ±b/a and c = 0. Substituting for m in y = mx + c we have as the equation for an asymptote y — ± (b/a)x. The combined equation for both asymptotes is given by: 1.5.8.9 Circle (c = 0) The circle may be regarded as a particular case of the ellipse (see above).
eBook - PDFSixth Form Pure Mathematics
Volume 1
- C. Plumpton, W. A. Tomkys(Authors)
- 2014(Publication Date)
- Pergamon(Publisher)
Show that the distance from the origin at which this tangent meets the line y = x tan / is independent of /. (N.) 56. Find the equations of the two circles which touch the lines x+2y = 1, x = 0, y = 0 and have their centres on the line JC = y. (O.C.) 57. The coordinates of a point on a plane curve are given parametrically by the relations x = a cos 3 /, y = a sin 3 /. Show that the equation of the tangent to the curve at the point whose parameter is / is given by x sin /-f y cos / = a sin / cos / and deduce that the length of the perpendicular from the point (a, 0) to this tangent has a maximum value 3a/3/4. (O.C.) 196 SIXTH FORM PURE MATHEMATICS 58. If f(x) is a function of x and/'(*) is its differential coefficient, show that, when h is sma.U,f(a+h) is approximately equal to fa) + hf '(a). Without using trigonometrical tables, find to three significant figures (i) the value of cos 31°, and (ii) the positive acute angle whose sine is 0.503. Give your answer to (ii) in degrees and tenths of a degree. (O.C.) 59. The centres of two circles of radii R and r are respectively (-b, 0) and (a, 0) where a and b are positive. The equation of the direct common tangent is x cos a +y sin a = a cos /?, where a and f$ are acute. Prove that R = acosp + b cos oc and show that, if R = 2r, a cos fi = (b + 2a) cos a. (O.C.)
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