Unit 6 Introduction to Trigonometry The Unit Circle (Unit 6.3) - PowerPoint PPT Presentation

Unit 6 – Introduction to Trigonometry The Unit Circle (Unit 6.3) William (Bill) Finch Mathematics Department Denton High School

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary Lesson Goals When you have completed this lesson you will: ◮ Find values of trigonometric functions for any angle. ◮ Find the values of trigonometric functions using the unit circle. W. Finch DHS Math Dept Unit Circle 2 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary Lesson Goals When you have completed this lesson you will: ◮ Find values of trigonometric functions for any angle. ◮ Find the values of trigonometric functions using the unit circle. W. Finch DHS Math Dept Unit Circle 2 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary Lesson Goals When you have completed this lesson you will: ◮ Find values of trigonometric functions for any angle. ◮ Find the values of trigonometric functions using the unit circle. W. Finch DHS Math Dept Unit Circle 2 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary Trigonometric Functions of Any Angle θ is an angle in standard position, y P ( x , y ) is a point on the terminal side, and r is the distance from P to P ( x , y ) θ the origin (denominators � = 0): r x sin θ = y csc θ = r r y Recall the equation of a cos θ = x sec θ = r circle centered at the r x origin: r 2 = x 2 + y 2 tan θ = y cot θ = x x y x 2 + y 2 � r = W. Finch DHS Math Dept Unit Circle 3 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary Example 1 The point ( − 3 , 2) is on the terminal side of an angle in standard position. Find the exact values of the six trigonometric functions of θ . W. Finch DHS Math Dept Unit Circle 4 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary Quadrental Angles Quadrental angles terminate on an axis. y y y y θ θ θ (0 , r ) θ x x x x ( r , 0) ( − r , 0) (0 , − r ) θ = 0 ◦ or θ = 90 ◦ or θ = 180 ◦ or θ = 270 ◦ or 0 radians π/ 2 radians π radians 3 π/ 2 radians W. Finch DHS Math Dept Unit Circle 5 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary Example 2 Find the exact value of each trigonometric function, if defined. If not defined, write undefined . a) cos π b) tan( − 270 ◦ ) c) sec 3 π 2 d) sin 5 π W. Finch DHS Math Dept Unit Circle 6 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary Angles Not Acute or Quadrental Quadrant II y sin θ = b sin θ ′ = b ( − a , b ) r r r b θ cos θ = − a cos θ ′ = a θ ′ x r r a tan θ = − b tan θ ′ = b a a W. Finch DHS Math Dept Unit Circle 7 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary Angles Not Acute or Quadrental Quadrant III y sin θ = − b sin θ ′ = b r r θ cos θ = − a cos θ ′ = a a x r r θ ′ tan θ = b tan θ ′ = b b r a a ( − a , − b ) W. Finch DHS Math Dept Unit Circle 8 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary Angles Not Acute or Quadrental Quadrant IV y sin θ = − b sin θ ′ = b r r θ cos θ = a cos θ ′ = a a x r r θ ′ tan θ = − b tan θ ′ = b b r a a ( a , − b ) W. Finch DHS Math Dept Unit Circle 9 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary Reference Angles If θ is an angle in standard position, its reference angle θ ′ is the acute angle formed by the terminal side of θ and the x -axis. y y y y θ θ θ θ ′ θ x x x x θ ′ θ ′ θ ′ = θ θ ′ = 180 ◦ − θ θ ′ = θ − 180 ◦ θ ′ = 360 ◦ − θ θ ′ = π − θ θ ′ = θ − π θ ′ = 2 π − θ W. Finch DHS Math Dept Unit Circle 10 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary Example 3 Sketch the angle and the identify its reference angle. a) − 150 ◦ b) 315 ◦ c) 3 π 4 d) 5 π 3 W. Finch DHS Math Dept Unit Circle 11 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary Evaluating Trigonometric Functions of Any Angle y 1. Sketch the angle. Quad II Quad I 2. Determine the reference angle θ ′ . sin θ : + sin θ : + cos θ : − cos θ : + 3. Find the value of the trig tan θ : − tan θ : + function for θ ′ . x Quad III Quad IV 4. Determine the sign (pos or neg) sin θ : − sin θ : − based on the quadrant cos θ : − cos θ : + tan θ : + tan θ : − containing the terminal side of θ . W. Finch DHS Math Dept Unit Circle 12 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary Special Reference Angles π π π θ (radians) 6 4 3 θ (degrees) 30 ◦ 45 ◦ 60 ◦ √ √ 1 2 3 sin θ 2 2 2 √ √ 3 2 1 cos θ 2 2 2 √ √ 3 tan θ 1 3 3 W. Finch DHS Math Dept Unit Circle 13 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary Example 4 Find the exact value of each expression. a) sin 4 π 3 b) sec 15 π 4 c) tan 150 ◦ d) cos ( − 120 ◦ ) W. Finch DHS Math Dept Unit Circle 14 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary Example 5 √ 29 Let sec θ = 5 , where sin θ > 0. Find the exact values of the remaining five trigonometric functions of θ . W. Finch DHS Math Dept Unit Circle 15 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary The Unit Circle A unit circle is a circle of y radius 1 centered at the origin. (0 , 1) The radian measure of a central angle is r s θ θ = s r = s x 1 = s r ( − 1 , 0) (1 , 0) so the arc length intercepted by θ equals the angle’s radian (0 , − 1) measure. W. Finch DHS Math Dept Unit Circle 16 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary The Unit Circle and the Wrapping Function Place a number line vertically y tangent to a unit circle at (1 , 0). P ( x , y ) Wrap this line around the circle t (counterclockwise for positive values t and clockwise for negative values), x each point t on the line would map to 1 (1 , 0) a unique point P ( x , y ) on the circle. This is referred to as the wrapping function w ( t ). Since r = 1, the six trigonometric rations of angle t can be defined in terms of just x and y . W. Finch DHS Math Dept Unit Circle 17 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary Trigonometric Functions on the Unit Circle y P ( x , y ) csc t = 1 sin t = y P (cos t , sin t ) y t sec t = 1 t cos t = x x x 1 tan t = y cot t = x x y And, of course, no These functions are referred to denominator = 0. as circular functions W. Finch DHS Math Dept Unit Circle 18 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary 16-Point Unit Circle y (0 , 1) √ √ � � � � − 1 2 , 3 2 , 1 3 2 2 � √ √ √ √ � � � 2 2 2 2 − 2 , 2 , 90 ◦ 2 2 120 ◦ 60 ◦ � √ √ � � � 2 , 1 3 2 , 1 3 − 2 135 ◦ 45 ◦ 2 π 2 π 2 π 150 ◦ 3 3 30 ◦ 3 π π 4 4 5 π π 6 6 ( − 1 , 0) (1 , 0) π x 180 ◦ 2 π 360 ◦ 0 ◦ 7 π 11 π 6 5 π 6 7 π 210 ◦ 4 4 330 ◦ 4 π 5 π 3 3 π 3 � √ 2 √ � � 225 ◦ 315 ◦ � 2 , − 1 3 2 , − 1 3 − 2 2 240 ◦ 300 ◦ � √ √ √ √ � � � 2 , − 2 2 270 ◦ 2 , − 2 2 − 2 2 √ √ � � � � − 1 3 1 3 2 , − 2 , − 2 2 (0 , − 1) W. Finch DHS Math Dept Unit Circle 19 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary 16-Point Unit Circle y (0 , 1) √ (0 , 1) √ � � � � − 1 3 1 3 2 , 2 , √ 2 � � 2 1 3 � √ √ √ 2 , √ � � � 2 2 2 2 − 2 , 2 , 2 90 ◦ 2 2 � √ √ 120 ◦ 60 ◦ � � √ √ 2 2 � � 2 , � 2 , 1 3 2 , 1 3 − ◦ 2 135 ◦ 45 ◦ 2 2 π 2 2 π 60 ◦ π � √ 150 ◦ 3 30 ◦ 3 3 π � π 2 , 1 3 4 4 5 π 45 ◦ π 2 6 6 ( − 1 , 0) (1 , 0) π π 180 ◦ 3 30 ◦ 2 π 360 ◦ 0 ◦ x π 4 π 7 π 11 π 6 6 5 π 6 7 π 210 ◦ 4 4 330 ◦ 4 π 5 π 3 3 π 3 (1 , 0) (1 , 0) 2 � √ √ � � 225 ◦ 315 ◦ � 2 , − 1 3 2 , − 1 3 x − 2 2 240 ◦ 300 ◦ � √ √ √ √ � � � 2 2 270 ◦ 2 2 − 2 , − 2 , − 2 2 √ √ � � � � − 1 3 1 3 2 , − 2 , − 2 2 (0 , − 1) W. Finch DHS Math Dept Unit Circle 20 / 25