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Math 170Trigonometry Trigonometry (from GreekΤριγωνομετρία "tri = three" + "gon = angle" + "metr[y] = to measure") is a branch of mathematics that deals with triangles, particularly those plane triangles in which one angle has 90 degrees (right triangles). Trigonometry deals with relationships between the sides and the angles of triangles and with the trigonometric functions, which describe those relationships. Trigonometry has applications in both pure mathematics and in applied mathematics, where it is essential in many branches of science and technology. It is usually taught in secondary schools either as a separate course or as part of a precalculus course.
Section 1.1 Angles Basic Terminology An angle consists of two rays in a plane with a common endpoint, or two segments with a common endpoint. A counterclockwise rotation generates a positive measure, while a clockwise rotation generates a negative measure.
Degree Measure The most common unit for measuring angles is the degree. This system was developed by the Babylonians, 4000 years ago. A complete rotation consists of 360 degrees. Hence one degree, written 1°, represents 1/360 of a complete rotation.
α β α β Complementary and Supplementary angles Two angles are said to be supplementary if their measures add up to 180°. Two angles are said to be complementary if their measures add up to 90°.
Minutes and Seconds. One degree is divided into 60 equal parts, each is called a minute, written 1’. One minute is divided into 60 equal parts, each is called a second, written 1”. Exercises a) 51°29’ + 32°46’ b) 90° – 73°12’
Converting between decimal degrees and Degrees, Minutes, and Seconds a) Convert 74°8’14” into decimal degrees to the nearest thousandth. b) Convert 34.817° into degrees, minutes, and seconds.
Standard Positions An angle is in standard position if its vertex is at the origin and its initial side lies on the positive x-axis.
Quadrantal Angles these are the angles with measures 90°, or 180°, or 270°, and so on. Coterminal Angles Two angles are said to be coterminal if they have the same initial side and same terminal side. The measures of any two coterminal angles differ by a (whole number) multiple of 360°.
Section 1.2 Angle Relationships and Similar Triangles Vertical angles Any pair of vertical angles have equal measure.
Parallel lines Two lines lying in the same plane are said to be parallel if they do not intersect no matter how far they are extended. Properties of parallel lines and angles.
Examples Find the measures of angles 1, 2, 3, and 4.
Angle sum of a triangle is always 180°. Cut out any size triangle from a piece of paper, tear out each corner from the triangle, you should be able to rearrange the pieces so that the three angles form a straight angle.
Similar Triangles are triangles of exactly the same shape but not necessarily the same size. Congruent Triangles Are triangles of exactly the same shape and same size. If two triangles are congruent, they must be similar. However, two similar triangles need not be congruent.
Conditions for similar triangles • Corresponding angles must have the same measure. • Corresponding sides must be proportional. (that is, the ratios of their corresponding sides must be equal.)
Examples Given that ∆ABC and ∆DFE are similar, find the lengths of the unknown sides of ∆DFE.
Section 1.4 Using the Definitions of the Trigonometric Functions Identities are equations that are true for all values of the variables for which all expressions are defined.
Caution ! Be sure not to use the inverse trigonometric function keys to find reciprocal function values. For example, sin-1(90°) is not equal to We will learn inverse trigonometric function in chapter 2.
