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## Angles and their Measures

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**Angles and their Measures**Lesson 1**As derived from the Greek Language, the word**trigonometrymeans “measurement of triangles.” • Initially, trigonometry dealt with relationships among the sides and angles of triangles and was used in the development of astronomy, navigation, and surveying.**With the development of Calculus and the physical sciences**in the 17th Century, a different perspective arose – one that viewed the classic trigonometric relationships as functions with the set of real numbers as their domain. • Consequently the applications expanded to include physical phenomena involving rotations and vibrations, including sound waves, light rays, planetary orbits, vibrating strings, pendulums, and orbits of atomic particles.**We will explore both perspectives beginning with angles and**their measures….. • An angle is determined by rotating a ray about its endpoint. • The starting position of called the initial side. The ending position is called the terminal side.**Standard PositionVertex is at the origin, and the initial**side is on the x-axis. Terminal Side Initial Side**Positive Angles are generated by counterclockwise rotation.**• Negative Angles are generated by clockwise rotation. • Let’s take a look at how negative angles are labeled on the coordinate graph.**Coterminal Angles**• Angles that have the same initial and terminal side. See the examples below.**Coterminal AnglesThey have the same initial and terminal**sides. Determine 2 coterminal angles, one positive and one negative for a 60 degree angle. 60 + 360 = 420 degrees 60 – 360 = -300 degrees**Give 2 coterminal angles.**30 + 360 = 390 degrees 30 – 360 = -330 degrees**Give a coterminal angle, one positive and one negative.**230 + 360 = 590 degrees 230 – 360 = -130 degrees**Give a coterminal angle, one positive and one negative.**-20 + 360 = 340 degrees -20 – 360 = -380 degrees**Give a coterminal angle, one positive and one negative.**Good but not best answer. 460 + 360 = 820 degrees 460 – 360 = 100 degrees 100 – 360 = -260 degrees**Complementary AnglesSum of the angles is 90**Find the complement of each angles: 40 + x = 90 x = 50 degrees No Complement!**Supplementary AnglesSum of the angles is 180**Find the supplement of each angles: 40 + x = 180 x = 140 degrees 120 + x = 180 x = 60 degrees**Coterminal Angles: To find a Complementary Angle:To find a**Supplementary Angle:**Radian Measure**• One radian is the measure of the central angle, , that intercepts an arc, s, that is equal in length to the radius r of the circle. • So…1 revolution is equal to 2π radians**Let’s take a**look at them placed on the unit circle.**Radians**Now, let’s add more…..**More Common Angles**Let’s take a look at more common angles that are found in the unit circle.**Determine the Quadrant of the terminal side of each given**angle. Q1 Go a little more than one quadrant – negative. Q3 A little more than one revolution. Q1**Determine the Quadrant of the terminal side of each given**angle. Q3 Q2 2 Rev + 280 degrees. Q4**Find a coterminal angle.**There are an infinite number of coterminal angles!**Find the complement & supplement of each angles, if**possible: None**Coterminal Angles: To find a Complementary Angle:To find a**Supplementary Angle: RECAP**Arc Length**Arc Length = (radius) (angle) The relationship between arc length, radius, and central angle is