Presentation on theme: "Let’s extend our knowledge of trigonometric functions…"— Presentation transcript:
1Let’s extend our knowledge of trigonometric functions… …in Section 4.3a
2Trigonometric Functions of Any Angle (not just positive or acute angles)A more “dynamic” view of angles:Initial Side – beginning position of a ray (part of the angle)Vertex – the endpoint of this rayTerminal Side – final position of a ray, after being rotated aboutthe vertexMeasure of an Angle – a number that describes the amount ofrotation from the initial side to the terminal side of the anglePositive angles are generated by counterclockwise rotationsand negative angles are generated by clockwise rotations.
3Trigonometric Functions of Any Angle (not just positive or acute angles)terminalsideterminalsideinitial sideinitial sideA positive angle instandard positionA negative angle instandard position
4Trigonometric Functions of Any Angle (not just positive or acute angles)Coterminal Angles – angles that have the same initial side andthe same terminal side, but have different measures.Positive and negativecoterminal anglesTwo positivecoterminal angles
5Guided PracticeFind and draw a positive angle and a negative angle that arecoterminal with the given angle.Note: there are infinitely many possible solutions…Here are two:(a)
6Guided PracticeFind and draw a positive angle and a negative angle that arecoterminal with the given angle.(b)The diagram?(c)The diagram? I believe that the diagram should always come first!
7Definition: Trigonometric Functions of any Angle Let be any angle in standard position and let P(x, y) be anypoint on the terminal side of the angle (except the origin). Let rdenote the distance from P to the origin ( ). ThenP(x,y)yrx
8Guided PracticeLet be the acute angle in standard position whose terminalside contains the point (5, 3). Find the six trig functions of .The diagram?P(5,3)35
9Guided PracticeLet be any angle in standard position whose terminal sidecontains the point (–5, 3). Find the six trig functions of .The diagram?P(–5,3)3–5
10Guided Practice Find the six trigonometric functions of . First, let’s draw a diagram, complete with a reference triangle:P(1,–1)
11Whiteboard PracticeFind each of the following without a calculator by using ratios ina reference triangle:(a)(b)(c)