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AccPreCalc Lesson 27

Essential Question: How are trigonometric equations solved?

Standards: Prove and apply trigonometric identities

MCC9‐12.F.TF.9 (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Solving Trig EquationsSolve: 2sin(Ѳ) -1 = 0

2Sin( Ѳ) = 1 Sin(Ѳ) = ½

Ѳ = ∏/6 , 5 ∏/6 , 13 ∏/6, ….

Restrict all solution from 0 to 2 ∏

Prove the following 1 + cos(2x) = cot(x) Sin(2x)

1 + cos2(x) – sin2(x) cos(x) / sin(x) 2sin(x) cos(x) cos2(x) + cos2(x) 2sin(x) cos(x)

2cos 2(x) 2sin(x) cos(x)

Cos(x) / sin(x) QED

Solve the equation

Sin(x) + sqrt(2) = -Sin(X)2sin(x) + sqrt(2) = 0 2sin(x) = -sqrt (2) Sin(x) = -sqrt(2) / 2 X = 5∏/4, 7∏/4

Check by plugging back into the equation

Solve the equation

3tan2(x) -1 = 0 Tan2(x) = 1/3 Tan(x) = +/_ 1/sqrt(3) rationalize your

answerTan(x) = +/- sqrt(3) / 3 X = ∏/6, 5∏/6, 7 ∏/6, 11 ∏/6

Solve for x

• Cot(x) cos2(x) = 2cot(x) Cot(x) cos2(x) - 2 cot(x) = 0 Cot(x) [cos2(x) -2] = 0 Set each factor equal to zero Cot(x) = 0 cos2(x) -2 = 0 X = ∏/ 2, cos2(x) = 23 ∏ /2 COS(X) = +/- SQRT(2)

No solutions

Solving multiple angle equations

• Sin(2x) = 1 2x = ∏/ 2 2x = 5 ∏/ 2 X = ∏/4 x = 5 ∏/4

2x = 9 ∏/2 X = 9 ∏/4 ….. This is bigger than 2 ∏ so this is

not one of our solutions.

Multiple angle equation

• Cos(3x) = - sqrt(3) / 2 3x = 5 ∏/6 3x = 7 ∏/6 3x = 31 ∏/6

3x = 17 ∏/6 3x = 19 ∏/6 3x = 29 ∏/6

Solve

• 2sin2(2x) = 1 on 0 to ∏• sin2(2x)= ½ • Sin(2x) = +/- sqrt(2) / 2 2x = ∏/4 2x = 3 ∏ /4 2x = 5 ∏ /4

2x = 7 ∏ /4 2x = 9 ∏ /4