6 3 Trig Equations Involving Multiple Angles 6

6. 3 Trig Equations Involving Multiple Angles If you tackle a trig equation with

6. 3 Trig Equations Involving Multiple Angles • Addition / Multiplication Property • Factoring

Ex 1 We see that cos 2θ – being positive – is in Q

Ex 2 Remember, tangent has a period of π, not 2π.

Ex 2 Remember, tangent has a period of π, not 2π. Tan = 1

Ex 2 If we wanted exact values between 0 and π, just let k

Ex 3 Do we recognize this “sum” formula?

Ex 4 Find ALL solutions in degrees. We notice that this is in quadratic

Ex 4 (2 sin 3θ + 1)(sin 3θ – 1) = 0 3θ =

Ex 5 Can we take the square roots of both sides?

Ex 6 In an earlier section, squaring both sides was a good idea. .

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6. 3 Trig Equations Involving Multiple Angles

6. 3 Trig Equations Involving Multiple Angles If you tackle a trig equation with 2θ or 3θ, that enters into your calculations. . . and you get more answers.

6. 3 Trig Equations Involving Multiple Angles • Addition / Multiplication Property • Factoring • Quadratic equations • Square both sides (E. S. )

Ex 1

Ex 1 We see that cos 2θ – being positive – is in Q I and Q IV.

Ex 1 We see that cos 2θ – being positive – is in Q I and Q IV. The reference angle is 30 o.

Ex 1 2θ = 30 + 360 k 2θ = 330 + 360 k We see that cos 2θ – being positive – is in Q I and Q IV. The reference angle is 30 o.

Ex 1 2θ = 30 + 360 k 2 2θ = 330 + 360 k 2 2 2

Ex 1 2θ = 30 + 360 k 2 2θ = 330 + 360 k 2 2 2 θ = 15 + 180 k θ = 165 + 180 k

Ex 1 2θ = 30 + 360 k 2 2θ = 330 + 360 k 2 2 2 θ = 15 + 180(0) θ = 165 + 180(0)

Ex 1 2θ = 30 + 360 k 2 2θ = 330 + 360 k 2 2 2 θ = 15 + 180(0) θ = 165 + 180(0) θ = 15, 165

Ex 1 2θ = 30 + 360 k 2 2θ = 330 + 360 k 2 2 2 θ = 15 + 180(1) θ = 165 + 180(1) θ = 15, 165

Ex 1 2θ = 30 + 360 k 2 2θ = 330 + 360 k 2 2 2 θ = 15 + 180(1) θ = 165 + 180(1) θ = 15, 165, 195, 345

Ex 2

Ex 2 Remember, tangent has a period of π, not 2π.

Ex 2 Remember, tangent has a period of π, not 2π. Tan = 1 when the angle is 45 o.

Ex 2

Ex 2

Ex 2 If we wanted exact values between 0 and π, just let k = 0, 1, and 2.

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Ex 3

Ex 3

Ex 3 Do we recognize this “sum” formula?

Ex 3

Ex 3

Ex 3

Ex 3

Ex 3

Ex 3

Ex 3 k=0

Ex 3 k=1

Ex 3 k=2

Ex 4

Ex 4 Find ALL solutions in degrees.

Ex 4 Find ALL solutions in degrees. We notice that this is in quadratic form.

Ex 4 (2 sin 3θ + 1)(sin 3θ – 1) = 0

Ex 4 (2 sin 3θ + 1)(sin 3θ – 1) = 0 3θ = 210 + 360 k, 330 + 360 k

Ex 4 (2 sin 3θ + 1)(sin 3θ – 1) = 0 3θ = 210 + 360 k, 330 + 360 k, 90 + 360 k

Ex 4 (2 sin 3θ + 1)(sin 3θ – 1) = 0 3θ = 210 + 360 k, 330 + 360 k, 90 + 360 k θ = 70 + 120 k, 110 + 120 k, 30 + 120 k

Ex 4 (2 sin 3θ + 1)(sin 3θ – 1) = 0 3θ = 210 + 360 k, 330 + 360 k, 90 + 360 k θ = 70 + 120 k, 110 + 120 k, 30 + 120 k 30, 70, 110, 150, 190, 230, 270, 310, 350

Ex 4 (2 sin 3θ + 1)(sin 3θ – 1) = 0 3θ = 210 + 360 k, 330 + 360 k, 90 + 360 k θ = 70 + 120 k, 110 + 120 k, 30 + 120 k 30, 70, 110, 150, 190, 230, 270, 310, 350 SHORTCUT: 30 o + 40 k