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Trigonometric Equations. In quadratic form, using identities or linear in sine and cosine. Solving a Trig Equation in Quadratic Form. Solve the equation: 2sin 2 θ – 3 sin θ + 1 = 0, 0 ≤ θ ≤ 2 p Let sin θ equal some variable sin θ = a Factor this equation (2a – 1) (a – 1) = 0 - PowerPoint PPT Presentation

### Transcript of Trigonometric Equations

• Trigonometric EquationsIn quadratic form, using identities or linear in sine and cosine

• Solving a Trig Equation in Quadratic FormSolve the equation:2sin2 3 sin + 1 = 0, 0 2pLet sin equal some variablesin = aFactor this equation(2a 1) (a 1) = 0Therefore a = a = 1

• Solving a Trig Equation in Quadratic FormNow substitute sin back in for asin = sin = 1Now do the inverse sin to find what equals = sin-1 () = sin-1 1 = p/6 and 5p/6 = p/2

• Solving a Trig Equation in Quadratic Form Solve the equation:(tan 1)(sec 1) = 0tan 1 = 0sec 1 = 0tan = 1sec = 1 = tan-1 1 = sec-1 1 = p/4 and 5p/4 = 0

• Solving a Trig Equation Using IdentitiesIn order to solve trig equations, we want to have a single trig word in the equation. We can use trig identities to accomplish this goal.Solve the equation3 cos + 3 = 2 sin2 Use the pythagorean identities to change sin2 to cos

• Solving a Trig Equation Using Identitiessin2 = 1 cos2 Substituting into the equation3 cos + 3 = 2(1 cos2 )To solve a quadratic equation it must be equal to 02cos2 + 3 cos + 1 = 0Let cos = b

• Solving a Trig Equation using Identities2b2 + 3b + 1 = 0(2b + 1) (b + 1) = 0(2b + 1) = 0b + 1 = 0b = -b = -1cos = -cos = -1 = 2p/3, 4p/3 = p

• Solving a Trig Equation Using Identitiescos2 sin2 + sin = 01 sin2 sin2 + sin = 0-2sin2 + sin + 1 = 02 sin2 sin 1 = 0Let c = sin 2c2 c 1 = 0(2c + 1) (c 1) = 0

• Solving a Trig Equation Using Identities(2c + 1) = 0c 1 = 0c = -c = 1sin = -sin = 1 = p/3 + p q=2p-p/3 = p/2 = 4p/3, q = 7p/3

• Solving a Trig Equation Using IdentitiesSolve the equationsin (2) sin = cos Substitute in the formula for sin 2(2sin cos )sin =cos 2sin2 cos cos = 0cos (2sin2 1) = 0cos = 02sin2 =1

• Solving a Trig Equation Using Identitiescos = 0

= 0, p = p/4, 3p/4, 5p/4, 7p/4

• Solving a Trig Equation Using Identitiessin cos = -This looks very much like the sin double angle formula. The only thing missing is the two in front of it.So . . . multiply both sides by 22 sin cos = -1 sin 2 = -12 = sin-1 -1

• Solving a Trig Equation Using Identities2 = 3p/2 = 3p/4 2 = 3p/2 + 2p 2q = 7p/2 q = 7p/4

• Solving a Trig Equation Linear in sin and cos sin + cos = 1There is nothing I can substitute in for in this problem. The best way to solve this equation is to force a pythagorean identity by squaring both sides.(sin + cos )2 = 12

• Solving a Trig Equation Linear in sin and cos sin2 + 2sin cos + cos2 = 12sin cos + 1 = 12sin cos = 0sin 2 = 02 = 02 = p = 0 = p/2 = p = 3p/2

• Solving a Trig Equation Linear in sin and cos Since we squared both sides, these answers may not all be correct (when you square a negative number it becomes positive).In the original equation, there were no terms that were squared

• Solving a Trig Equation Linear in sin and cos Check:Does sin 0 + cos 0 = 1?Does sin p/2 + cos p/2 = 1?Does sin p + cos p = 1?Does sin 3p/2 + cos 3p/2 = 1?

• Solving a Trig Equation Linear in sin and cos sec = tan + cot sec2 = (tan + cot )2sec2 = tan2 + 2 tan cot + cot2 sec2 = tan2 + 2 + cot2 sec2 tan2 = 2 + cot2 1 = 2 + cot2 -1 = cot2

• Solving a Trig Equation Linear in sin and cos q is undefined (cant take the square root of a negative number).

• Solving Trig Equations Using a Graphing UtilitySolve 5 sin x + x = 3. Express the solution(s) rounded to two decimal places.Put 5 sin x + x on y1Put 3 on y2Graph using the window 0 2pFind the intersection point(s)

• Word ProblemsPage 519 problem 58