7/9/ : Extending the Trig Ratios Expectation: G1.3.3: Determine the exact values of sine, cosine, and tangent for 0°, 30°, 45°, 60°, and their integer multiples and apply in various contexts.
If the angles ∠ X and ∠ Y each measure between 0° and 90°, and if sin X = cos Y, what is the sum of the measures of the angles ∠ X and ∠ Y? A. 30 B. 45 C. 60 D. 90 E /9/ : Extending the Trig Ratios
7/9/ : Extending the Trig Ratios Angle of Rotation An angle is an angle of rotation iff: a. its vertex is the origin b. one side is the positive x-axis c. the other side is a rotation of the first side centered at the origin.
7/9/ : Extending the Trig Ratios Angles of Rotation θ
7/9/ : Extending the Trig Ratios Angles of Rotation θ
7/9/ : Extending the Trig Ratios Angles of Rotation θ
7/9/ : Extending the Trig Ratios Angles of Rotation θ
7/9/ : Extending the Trig Ratios Unit Circle Defn: A circle is a unit circle iff: a. its center is the origin (0,0). b. its radius is 1.
7/9/ : Extending the Trig Ratios Unit Circle: x 2 + y 2 = 1 (1,0) (0,-1) (0,1) (-1,0)
7/9/ : Extending the Trig Ratios Who Cares? We can use unit circles and trig to find coordinates of points on a unit circle.
7/9/ : Extending the Trig Ratios (1,0) (0,-1) (0,1) (-1,0) 30° A What are the coordinates of A?
7/9/ : Extending the Trig Ratios What are the coordinates of B? (1,0) (0,-1) (0,1) (-1,0) 45° B
7/9/ : Extending the Trig Ratios What are the coordinates of C? (1,0) (0,-1) (0,1) (-1,0) C 60°
7/9/ : Extending the Trig Ratios What are the coordinates of D? (1,0) (0,-1) (0,1) (-1,0) 60° D
7/9/ : Extending the Trig Ratios What is the angle of rotation for the hypotenuse below? (1,0) (0,-1) (0,1) (-1,0) A (.866,.5)
7/9/ : Extending the Trig Ratios ??????????? What is the cos 30? What is the sin 30? Compare sin 30, cos 30 and the (x,y) coordinates of A.
7/9/ : Extending the Trig Ratios What is the angle of rotation for the hypotenuse below? (1,0) (0,-1) (0,1) (-1,0) B (-.707,.707)
7/9/ : Extending the Trig Ratios ??????????? What is the cos 135? What is the sin 135? Compare sin 135, cos 135 and the (x,y) coordinates of B.
7/9/ : Extending the Trig Ratios What is the angle of rotation for the hypotenuse below? (1,0) (0,-1) (0,1) (-1,0) C (-.5, -.866)
7/9/ : Extending the Trig Ratios ??????????? What is the cos 240? What is the sin 240? Compare sin 240, cos 240 and the (x,y) coordinates of C.
7/9/ : Extending the Trig Ratios What is the angle of rotation for the hypotenuse below? (1,0) (0,-1) (0,1) (-1,0) D (.5, -.866)
7/9/ : Extending the Trig Ratios ??????????? What is the cos 300? What is the sin 300? Compare sin 300, cos 300 and the (x,y) coordinates of D.
7/9/ : Extending the Trig Ratios Sine and Cosine on a Unit Circle Defn: Let θ be a rotation angle. Then sin θ is the y-coordinate of the image of P(1,0) rotated θ about the origin and cos θ is the x-coordinate. P’= (cos θ, sin θ)
7/9/ : Extending the Trig Ratios What are sin (-60) and cos (-60)? What are the sin 440 and cos 440?
7/9/ : Extending the Trig Ratios Negative Angles sin (- θ ) = - sin θ cos (- θ ) = cos ( θ )
7/9/ : Extending the Trig Ratios Angles Larger than 360° If θ > 360, then: sin θ = sin ( θ - 360n) cos θ = cos ( θ - 360n) where n is a whole number.
7/9/ : Extending the Trig Ratios Verify trig identity number 1: tan θ = sin θ cos θ
7/9/ : Extending the Trig Ratios Verify trig identity number 2: sin 2 θ + cos 2 θ = 1
7/9/ : Extending the Trig Ratios Graphing Sine and Cosine For θ = 0, 30, 45, 60, 90, 120, 135, 150, 180, 210, 225, 240, 270, 300, 315, 330 and 360, determine sin θ and cos θ. It may be helpful to organize your data into a chart. Graph your data.
7/9/ : Extending the Trig Ratios
7/9/ : Extending the Trig Ratios A satellite orbits a planet at 1° per hour. Let the radius of the orbit equal 1 and determine the ordered pair coordinates of the satellite after 497 hours. Assume it starts at (1,0).
7/9/ : Extending the Trig Ratios Give 2 angles, θ, between 0 and 360 that have cos θ =.7071.
7/9/ : Extending the Trig Ratios Assignment pages , # (odds)