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Graphs of other Trig Functions

Explore the essential concepts of cosecant, secant, tangent, and cotangent functions within the context of trigonometric graphs. This section covers the definitions, key characteristics, and specific points of these functions, emphasizing where they are undefined due to sine and cosine values being zero. Learn to sketch the cosecant and secant curves with vertical asymptotes, and understand the relationships between tangent and cotangent curves, including their own asymptotes. This comprehensive overview aids in grasping the fundamental behavior of these trigonometric functions.

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Graphs of other Trig Functions

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  1. Graphs of other Trig Functions Section 4.6

  2. What is the cosecant x? • Where is cosecant not defined? • Any place that the Sin x = 0 • The curve will not pass through these points on the x-axis. x = 0, π, 2 π Cosecant Curve

  3. Drawing the cosecant curve • Draw the reciprocal curve • Add vertical asymptotes wherever curve goes through horizontal axis • “Hills” become “Valleys” and “Valleys” become “Hills” Cosecant Curve

  4. y = Csc x → y = Sin x 1 -1 Cosecant Curve

  5. y = 3 Csc (4x – π) → y = 3 Sin (4x – π) c = π a = 3 b = 4 Per. = P.S. = dis. = 3 -3 Cosecant Curve

  6. y = -2 Csc 4x + 2 → y = -2 Sin 4x + 2 4 2 Cosecant Curve

  7. What is the secant x? • Where is secant not defined? • Any place that the Cos x = 0 • The curve will not pass through these points on the x-axis. Secant Curve

  8. y = Sec 2x → y = Cos 2x 1 -1 Secant Curve

  9. y = Sec x → y = Cos x 1 -1 Secant Curve

  10. y = 3 Csc (πx – 2π) • y = 2 Sec (x + ) • y = ½ Csc (x - ) • y = -2 Sec (4x + 2) Graph these curves

  11. y = 3Csc (πx – 2π) → y = 3 Sin (π x – 2π) 3 -3

  12. y = 2Sec (x + ) → y = 2 Cos (x + ) 2 -2

  13. y = ½ Csc (x - ) → y = ½ Csc (x - ) ½ - ½

  14. y = -2 Sec (4π x + 2 π) -2 Cos (4π x + 2 π) 2 -2

  15. Graph of Tangent and Cotangent Still section 4.6

  16. Define tangent in terms of sine and cosine • Where is tangent undefined? Tangent

  17. y = Tan x

  18. So far, we have the curve and 3 key points • Last two key points come from the midpoints between our asymptotes and the midpoint • Between and 0 and between and 0 • → and Tangent Curve

  19. y = Tan x x 0 y =Tan x und. -1 0 1 und. 1 -1

  20. For variations of the tangent curve • Asymptotes are found by using: A1. bx – c = A2. bx – c = • Midpt. = • Key Pts: and

  21. y = 2Tan 2x x y =2Tan 2x und. und. bx – c = bx – c = 2x = 2x= x = x =

  22. y = 2Tan 2x x 0 y =2Tan 2x und. -2 0 2 und. = 0 Midpt = K.P. = = K.P. = =

  23. y = 4Tan x 0 y =4Tan und. -4 0 4 und.

  24. y = 4Tan x 0 y =4Tan und. -4 0 4 und.

  25. Cotangent curve is very similar to the tangent curve. Only difference is asymptotes bx – c = 0 bx – c = π → 0 and π are where Cot is undefined Cotangent Curve

  26. y = 2Cot x π und. 2 0 -2 und. 2Cot

  27. y = 2Cot x π 2Cot und. 2 0 -2 und.

  28. y = 3Cot x 3Cot und. 3 0 -3 und.

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