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ANALYTIC TRIG!!!!!!!

ANALYTIC TRIG!!!!!!!. Trig Identities. sin^2x + cos^2x = 1 tan^2x + 1 = sec^2x 1 + cot^2x = csc^2x sin(-x) = -sinx cos(-x) = cosx tan(-x) = -tanx. Proving Trig Identities. Start with one side Use known identities Convert to sines and cosines. Example: Prove:

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ANALYTIC TRIG!!!!!!!

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  1. ANALYTIC TRIG!!!!!!!

  2. Trig Identities sin^2x + cos^2x = 1 tan^2x + 1 = sec^2x 1 + cot^2x = csc^2x sin(-x) = -sinx cos(-x) = cosx tan(-x) = -tanx

  3. Proving Trig Identities • Start with one side • Use known identities • Convert to sines and cosines

  4. Example: Prove: 2tanxsecx = 1/(1-sinx) – 1/(1+sinx)

  5. Solving Trig Equations Example: Tan^2x - 3 = 0

  6. Double-Angle Formulas Sine: sin2x = 2sinxcosx Cosine: cos2x = cos^2x – sin^2x Tangent: tan2x = (2tanx)/(1 – tan^2x)

  7. Example: If cosx = -2/3 and x is in quadrant II, find sin2x.

  8. Half-Angle Formulas Sinu/2 = +/- √(1-cosu)/2 Cosu/2 = +/- √(1+cosu)/2 Tanu/2 = (1-cosu)/sinu = sinu/(1+cosu)

  9. Example: Find the exact value of sin22.5°

  10. Vectors Component Form: y v = <x2 – x1, y2 – y1> Magnitude of a Vector: x |v| = √a^2 + b^2

  11. Algebraic Operations on Vectors If u = <a1, b1> and v = <a2, b2>, then u + v = <a1 + a2, b1+b2> cu = <ca1, cb1>

  12. Example: If u = <2,-3> and v = <-1,2> find u + v and -3v. u + v = <2 – 1, -3 + 2> = <1,-1> -3v = <-3(-1), -3(2)> = <3, -6>

  13. Vectors in terms of i and j i = <1, 0> j = <0,1> v = <a, b> = ai +bj u = <5, -8> u = 5i – 8j

  14. Horizontal and Vertical Components of a Vector Let v be a vector with magnitude |v| and direction θ. v = <a,b> = ai + bj a = vcosθ and b = vsinθ

  15. Example: An airplane heads due north at 300 mi/h. It experiences a 40 mi/h crosswind flowing in the direction N 30° E. v u

  16. a) Express the velocity v of the airplane relative to the air, and the velocity u of the wind in component form. v = 0i + 300j = 300j u = (40cos60°)i + (40sin60°)j u = 20i + 20√3j u = 20i + 34.64j

  17. b) Find the true velocity of the airplane as a vector. w = u + v w = (20i + 20√3j) + (300j) w = 20i + (20√3 + 300)j w = 20i + 334.64j

  18. c) Find the true speed of the airplane. Speed: w = √20^2 + 334.64^2 = 335.2 mi/h

  19. ADDITION AND SUBTRACTION FORMULAS

  20. SINE • Sin(s+t) = sinscost + cosssint • Sin(s-t) = sinscost – cosssint

  21. COSINE • Cos(s+t) = cosscost – sinssint • Cos(s-t) = cosscost + sinssint

  22. TANGENT • Tan(s+t) = (tans + tant)/(1 – tanstant) • Tan(s-t) = (tans – tant)/(1 + tanstant)

  23. INVERSETRIGONOMETRIC FUNCTIONS

  24. Domain: [-1, 1] Range: [-pi/2, pi/2]

  25. Domain: [-1, 1] Range: [0, pi]

  26. Domain: all real numbers Range: (-pi/2, pi/2)

  27. COMPLEX NUMBERS… THE TRIGONMETRIC VARIETY z = r(cos + isin ) • … where r = modulus of z = √a2 + b2

  28. DOT PRODUCTS u = (a1, b1) v = (a2, b2) u • v = a1a2 + b1b2

  29. DOTPRODUCTTHEOREM u • v = |u||v|cosθ

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