Precalc 8'3

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Precalc 8.3

• Algebraic Solutions to Trigonometric Equations

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We already know how to solve basic trig
equations.

• Examples
• Solve cos x 0.6 on -p, p
• x cos-1 (0.6) 0.9273
• Since cos(-x) cos(x), -0.9273 must also be
  answer.

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Lets beef it up a little.

• Example solve 8 cos x - 1 0 on -8, 8
• cos x 1/8
• x cos-1 (1/8) -1.4455
• Therefore, 1.4455 must also be answer.
• x 1.4455 2pn and x -1.4455 2pn

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Solving special equations.

• Sometimes the exact answer is very knowable. So
  well want you to know it. And share it. And
  write it.
• Example solve sin u v2/2
• Ah! u p/4.
• But sine is also positive in the second quadrant.
  So 3p/4 must also be an answer.
• u p/4 2pn and u 3p/4 2pn

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Substitutions can make you seem clever.

• Example Solve sin 2x 0.5.
• Make the famous u-substitution.
• Let u 2x.
• sin u 0.5 ---gt u p/6 and u 5p/6
• Now get it back in terms of x!
• ? 2x p/6 and 2x 5p/6
• ? x p/12 and x 5p/12
• Notice that the period is now p.
• ? x p/12 pn and x 5p/12 pn

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More substitution fun.

• Example Solve 3 sin2 x - sin x - 2 0 on -p,
  p
• Does this look familiar?
• Hint Let u sin x.
• Quadratic equation, baby!
• 3 u2 - u - 2 0 ---gt (3u 2)(u - 1)
• So u -2/3 or u 1
• ? sin x -2/3 or sin x 1.
• sin x -2/3 ---gt
• x -0.7297 and x p - (-0.7297) 3.8713.
• Dang, 3.8713 is outside -p, p BUT it IS
  coterminal with -2.4419 which IS inside the
  interval. Yay!
• sin x 1 ---gt x p/2.
• Final Answers x -0.7297, x -2.4419, x p/2

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Using Factoring.

• Solve tan x cos2 x tan x.
• tan x cos2 x - tan x 0
• (tan x)(cos2 x - 1) 0
• tan x 0 or cos2 x - 1 0
• tan x 0 ---gt x 0 x p
• ? x pn since tan has a period of p.
• cos2 x - 1 0 ---gt cos2 x 1
• cos x 1.
• x 0 x p
• ? x pn

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Use those identities when they appear!

• Solve -10 cos2 x - 3 sin x 9 0
• We dont like multiple trig functions floating
  around.
• -10(1 - sin2 x) - 3 sin x 9 0
• -10 10 sin2 x - 3 sin x 9 0
• 10 sin2 x - 3 sin x - 1 0
• (2 sin x - 1)(5 sin x 1) 0
• sin x 1/2 or sin x -1/5
• x p/6 and 5p/6 or x -0.2014 and 3.343
• x p/6 2pn x -0.2014 2pn
• x 5p/6 2pn or x 3.343 2pn

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When all failsquadratic formula.

• Solve sec2 x 5 tan x -2
• sec2 x 5 tan x 2 0
• (1 tan2 x) 5 tan x 2 0
• tan2 x 5 tan x 3 0
• Doesnt factor! Weak!
• tan x -0.6972 or tan x -4.3028
• x tan-1 (-0.6972) x tan-1 (-4.3028)
• x -.6088 pn x -1.3424 pn