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Reference Angles And Trigonometry Using Trigonometry in a Right Triangle We were limited to Acute Angles We can extend Trigonometry to Angles of Any.

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Presentation on theme: "Reference Angles And Trigonometry Using Trigonometry in a Right Triangle We were limited to Acute Angles We can extend Trigonometry to Angles of Any."— Presentation transcript:

1

2 Reference Angles And Trigonometry

3 Using Trigonometry in a Right Triangle We were limited to Acute Angles We can extend Trigonometry to Angles of Any Measure by placing those angles in the coordinate plane We do this by using reference angles, Acute Angles measured to the x-axis.

4 Angles are Placed with one side called the initial side on the positive x-axis. The terminal side is rotated counter-clockwise. 135 °

5 A Reference Angle is measured to the x-axis. The terminal side is rotated counter-clockwise. 135 ° 45 °

6 A Reference Angle is measured to the x-axis. The terminal side is rotated counter-clockwise. 225 ° 45 °

7 A Reference Angle is measured to the x-axis. The terminal side is rotated counter-clockwise. 315 ° 45 °

8 A Reference Angle is measured to the x-axis. If the terminal side is rotated clockwise, the angle measure is Negative. -45 ° 45 ° Always Positive.

9 Unit Circle has a radius of 1 unit.

10 Unit Circle has a radius of 1 unit. 45 ° 1 Cos + Sin + x= =y Cosine = x Sine = y

11 45 ° Cos + Sin ° Reference Angle = Cos - Sin +

12 45 ° Cos + Sin ° Reference Angle = Cos - Sin + Cos - Sin - 45 ° 135 ° 45 ° 225 °

13 45 ° Cos + Sin ° Reference Angle Cos - Sin + Cos - Sin - 45 ° 135 ° 45 ° 225 ° 45 ° Cos + Sin °

14 45 ° Cos + Sin + Cos - Sin + Cos - Sin - 45 ° 135 ° 45 ° 225 ° 45 ° Cos + Sin ° Quadrant 2 Quadrant 1 Quadrant 4 Quadrant 3

15 45 ° Cos + Sin + 45 ° Quadrant 1 Cosine = x Sine = y Tangent = Δy Δx Tangent = Sine Cosine tan = 1

16 45 ° Cos + Sin + Cos - Sin + Cos - Sin - 45 ° 135 ° 45 ° 225 ° 45 ° Cos + Sin ° Quadrant 2 Quadrant 1 Quadrant 4 Quadrant 3 tan = 1 tan = -1 tan = 1

17 Cos + Sin + Cos - Sin + Cos - Sin - Cos + Sin - Quadrant 2 Quadrant 1 Quadrant 4 Quadrant 3 Tan - Tan + Tangent = Sine Cosine

18 30 ° Cos + Sin + 30 ° 1 Cosine = x Sine = y

19 30 ° Cos + Sin + Cos - Sin + Cos - Sin - 30 ° 150 ° 30 ° 210 ° 30 ° Cos + Sin ° 150 °

20 Cos + Sin + Cos - Sin + Cos - Sin ° 30 ° 210 ° Cos + Sin ° Tangent = Sine Cosine

21 60 ° Cos + Sin + 60 ° 1

22 Cos + Sin + 60 ° 120 ° Cos + Sin °

23 60 ° Cos + Sin + 60 ° Cos - Sin ° 60 ° Cos - Sin °

24 60 ° Cos + Sin + 60 ° Cos - Sin ° 60 ° Cos - Sin ° 60 ° Cos + Sin °

25 Cos + Sin + 60 ° Cos - Sin ° Cos - Sin ° Cos + Sin ° Tangent = Sine Cosine

26 Cosine = x Sine = y Cos + Sin (1, 0) (-1, 0) (0, 1) (0, -1) Cos(0) =1 Sin(0) = 0 Cos Sin + Cos(90) =0 Sin(90) = 1 90 ° 180 ° Cos - Sin Cos(180) = -1 Sin(180) = ° Cos Sin - Cos(270) = 0 Sin(270) = -1

27 Cosine = x Sine = y Tangent = Sine Cosine Cos + Sin (1, 0) (-1, 0) (0, 1) (0, -1) Cos(0) =1 Sin(0) = 0 Cos Sin + Cos(90) =0 Sin(90) = 1 90 ° 180 ° Cos - Sin Cos(180) = -1 Sin(180) = ° Cos Sin - Cos(270) = 0 Sin(270) = -1 Tan(0) =0 Tan(90) undefined Tan(180) =0 Tan(270) undefined

28 Evaluate the trigonometric functions at each real number. = y = x

29 Evaluate the six trigonometric functions at each real number. (0, -1) = y = x = -1 = 0 DNE Does Not Exist DNE = -1 = 0

30 Evaluate the six trigonometric functions at each real number. Sin Cos Tan Csc Sec Cot So, you think you got it now?


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