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Trig Functions of Any Angle

Trig Functions of Any Angle. Section 4.4. θ. In first section, we calculated trig functions for acute angles. In this section, we are going to extend these basic definitions to cover any angle. θ. = 5. Plot the point (-3,4) Label the hypotenuse r and find its length. 5. r. 4. θ.

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Trig Functions of Any Angle

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  1. Trig Functions of Any Angle Section 4.4

  2. θ • In first section, we calculated trig functions for acute angles. • In this section, we are going to extend these basic definitions to cover any angle. θ

  3. = 5 • Plot the point (-3,4) • Label the hypotenuse r and find its length. 5 r 4 θ Sin θ = -3 Cos θ = Tan θ =

  4. Definitions of Trig Functions of Any Angle • Let θ be an angle in standard position with (x,y) a point on the terminal side. Then: Sin θ = Cos θ = Tan θ = Cscθ = Sec θ = Cot θ =

  5. Find the 6 trig functions of θ given that the ray ends at the point (-15, -8) Sin θ = Cos θ = Tan θ = Cscθ = Sec θ = Cot θ = -15 -8 17

  6. Find the 6 trig functions of θ given that the ray ends at the point (12, -5) Sin θ = Cos θ = Tan θ = Cscθ = Sec θ = Cot θ = 12 -5 13

  7. Quadrants • In which quadrants was the Sine positive? • I and II • In which quadrants was the Cosine positive? • I and IV • In which quadrants was the Tangent positive? • I and III

  8. Quadrants Sine is positive All Trig Functions are positive Students All Tangent is positive Cosine is positive Take Calculus

  9. → II • What quadrant is θ in if: • Sin θ > 0 and Cos θ < 0 • Tan θ > 0 and Cos θ < 0 • Sin θ < 0 and Tan θ < 0 • Cos θ > 0 and Tan θ > 0 → III → IV → I

  10. Given that Tan θ = - and Sin θ > 0, find the remaining 5 trig functions of θ. What quadrant? II 25 Sin θ = Cos θ = Tan θ = Cscθ = Sec θ = Cot θ = 7 -24

  11. Given that Cos θ = - and Sin θ < 0, find the remaining 5 trig functions of θ. What quadrant? III Sin θ = Cos θ = Tan θ = Cscθ = Sec θ = Cot θ = -4 -3 5

  12. Given that Sin θ = - and Tan θ < 0, find the remaining 5 trig functions of θ. What quadrant? IV Sin θ = Cos θ = Tan θ = Cscθ = Sec θ = Cot θ = 8 -15 17

  13. What did we learn • How to find the trig functions of an angle given a point on its terminal side • How to determine the quadrant of an angle based on trig functions • How to find the trig functions based on one function and criteria • Homework: Page 297, 1-24 odd

  14. Find the Sin, Cos, and Tan trig functions of θ given that the ray ends at the point (5,0) Sin θ = Cos θ = Tan θ = y = 0 5

  15. Quadrant Angles • On our Cartesian plane, we have 5 critical points: Find the Sine of these 5 angles -1 0 Sin = Sin 0 = Sin = 1 0 Sin 2π = 0 Sin π =

  16. Graph of the Sine Curve • Using these 5 points, we can create the Sine Curve 0

  17. Quadrant Angles • Using the same process, find the Cos of the 5 critical points. 0 1 Cos = Cos 0 = Cos = 0 1 Cos 2π = -1 Cos π =

  18. Graph of the Cosine Curve • Using these 5 points, we can create the Sine Curve 0

  19. Tan Values • Sin 0 = 0 Cos 0 = 1 Tan 0 = 0 • Sin = 1 Cos = 0 Tan = undef. • Sin π = 0 Cos π = -1 Tan π = 0 • Sin = -1 Cos = 0 Tan = undef. • Sin 2π = 0 Cos 2π = 1 Tan 2π = 0

  20. Reference Angles • The acute angle formed by the terminal side of an angle and the horizontal axis. • For an angle θ, we use θ’ to denote the reference angle

  21. Reference Angles • What is the reference angle for 210º Where is there an acute angle between the terminal side of the angle and the horizontal axis? θ’ = 210 – 180 = 30º

  22. Reference Angles • Find the reference angles for the following: • 330º • 225º • -225º • 750º = 360º - 330º = 30º = 225º - 180º = 45º = -180º - -225º = 45º = 750º - 720º = 30º

  23. Reference Angles • In general, for any angle θ θ’ = 180 - θ θ’ = θ θ’ = π - θ θ’ = θ - 180 θ’ = 360 - θ θ’ = θ - π θ’ = 2π - θ

  24. Reference Angles • Find the reference angle for 2nd Quadrant: → π – θ = π – =

  25. Reference Angles • So far, all we have been finding are reference angles. • We use reference angles to find the exact value of angles that are not acute. • We will use this for the remainder of the year. “GTK” – Good to Know

  26. Finding the Exact Value • Find the reference angle • Find the trig function of the reference angle • Check the sign of the function

  27. Sin 210º = - ½ 1. Find the reference angle Quadrant III 200º - 180º = 30º 2. Find the Sin of the reference angle Sin 30º = 3. Is it positive or negative? Negative ½

  28. Cos 330º 1. Find the reference angle Quadrant IV 360º - 330º = 30º 2. Find the Sin of the reference angle Cos 30º = 3. Is it positive or negative? Positive

  29. Find the Sin, Cos, and Tan of 135º 45º • Reference Angle = • Quadrant = • Sin 135º = • Cos 135º = • Tan 135º = II -1

  30. Find the Sin, Cos, and Tan of -240º 60º • Reference Angle = • Quadrant = • Sin -240º = • Cos -240º = • Tan -240º = II

  31. Find the Sin, Cos, and Tan of • Reference Angle = • Quadrant = • Sin = • Cos = • Tan = IV

  32. Find the: • Sin • Csc • Tan • Csc • Cot

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