Geometry and Trigonometry

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Geometry and Trigonometry The names sine, secant and tangent are derived from the geometric terms associated with certain lines in the diagram below. If the radius of the circle is taken to be one unit, it follows that CN = sinθ, or CN = sinθ 1 AB = tanθ, or AB = tanθ 1 OB = secθ, or OB = secθ 1 The geometric narratives for these lines are- • The sine of an arc is a straight line drawn from one end of that arc, perpendicular to a diameter passing through the other end of the same arc. • The tangent of an arc is a straight line drawn from one extremity, perpendicular to the diameter, and terminated by a straight line drawn through the center of the circle and the other extremity of that arc. • The secant of an arc is a straight line drawn from the center of the circle, and produced till it meets the tangent. Now CN is half of the chord CD, and early expressions for the sines of angles were really lengths of the chords in a circle. The Arabic word for chord was translated into the Latin sinus. The lengths AB and OB are portions of the tangent and the secant to the circle. The trigonometric functions bearing these names were developed later than the sine, and they merely assumed the corresponding geometric names. The cofunctions received their names from the fact that any trigonometric function of an acute angle is equal in value to the cofunction of the complementary angle. There are other(old) functions that can also be reckoned. The versed sine or versine of an angle (or arc) is that part of the diameter between the sine and the arc i.e., versθ = NA = 1 − cosθ. The coversed sine or coversine is the versine of the complement of an angle (or arc). Here coversθ = EF = 1− sinθ . The haversine is one-half of the versine. 1 havθ = (1− cosθ). 2.

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Trigonometric Functions
Trigonometric Functions This worksheet covers the basic characteristics of the sine, cosine, tangent, cotangent, secant, and cosecant trigonometric functions. Sine Function: f(x) = sin (x) • Graph • Domain: all real numbers • Range: [-1 , 1] • Period = 2π • x intercepts: x = kπ , where k is an integer. • y intercepts: y = 0 • Maximum points: (π/2 + 2kπ, 1), where k is an integer. • Minimum points: (3π/2 + 2kπ, -1), where k is an integer. • Symmetry: since sin (–x) = –sin (x) then sin(x) is an odd function and its graph is symmetric with respect to the origin (0, 0). • Intervals of increase/decrease: over one period and from 0 to 2π, sin (x) is increasing on the intervals (0, π/2) and (3π/2 , 2π), and decreasing on the interval (π/2 , 3π/2). Tutoring and Learning Centre, George Brown College 2014 www.georgebrown.ca/tlc Trigonometric Functions Cosine Function: f(x) = cos (x) • Graph • Domain: all real numbers • Range: [–1 , 1] • Period = 2π • x intercepts: x = π/2 + k π , where k is an integer. • y intercepts: y = 1 • Maximum points: (2 k π , 1) , where k is an integer. • Minimum points: (π + 2 k π , –1) , where k is an integer. • Symmetry: since cos(–x) = cos(x) then cos (x) is an even function and its graph is symmetric with respect to the y axis. • Intervals of increase/decrease: over one period and from 0 to 2π, cos (x) is decreasing on (0 , π) increasing on (π , 2π). Tutoring and Learning Centre, George Brown College 2014 www.georgebrown.ca/tlc Trigonometric Functions Tangent Function : f(x) = tan (x) • Graph • Domain: all real numbers except π/2 + k π, k is an integer.
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Coverrailway Curves Book.Cdr
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Inverse Trig Functions Summary
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Copyrighted Material
INDEX Abel, Niels Henrik, 234 algebraic manipulation, of integrand, finding by numerical methods, 367 abscissa, Web-H1 412 and improper integrals, 473 absolute convergence, 556, 557 alternating current, 324 as a line integral, 981 ratio test for, 558, 559 alternating harmonic series, 554 parametric curves, 369, 610 absolute error, 46 alternating series, 553–556 polar curve, 632 Euler’s Method, 501 alternating series test, 553, 560, 585 from vector viewpoint, 760, 761 absolute extrema amplitude arc length parametrization, 761, 762 Extreme-Value Theorem, 205 alternating current, 324 finding, 763, 764 finding on closed and bounded sets, simple harmonic motion, 124, properties, 765, 766 876–879 Web-P5 arccosine, 57 finding on finite closed intervals, 205, sin x and cos x, A21, Web-D7 Archimedean spiral, 626, 629 206 analytic geometry, 213, Web-F3 Archimedes, 253, 255 finding on infinite intervals, 206, 207 Anderson, Paul, 376 palimpsest, 255 functions with one relative extrema, angle(s), A1–A2, Web-A1–Web-A2 arcsecant 57 208, 209 finding from trigonometric functions, arcsine, 57 on open intervals, 207, 208 Web-A10 arctangent, 57 absolute extremum, 204, 872 of inclination, A7, Web-A10 area absolute maximum, 204, 872 between planes, 719, 720 antiderivative method, 256, 257 absolute minimum, 204, 872 polar, 618 calculated as double integral, 906, 907 absolute minimum values, 872 rectangular coordinate system, computing exact value of, 279 absolute value, Web-G1 A3–A4, Web-A3–Web-A4 definition, 277 and square roots, Web-G1, Web-B2 standard position,
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Development of Trigonometry Medieval Times Serge G
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