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Double-Angle and Half-Angle Formulas . Section 5.5. Multiple-Angle Formulas. In the previous sections, we used: The Fundamental Identities Sin²x + Cos²x = 1 Sum & Difference Formulas Cos (u – v) = Cos u Cos v + Sin u Sin v Now we will use double angle and half angle formulas.

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## Double-Angle and Half-Angle Formulas

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**Double-Angle and Half-Angle Formulas**Section 5.5**Multiple-Angle Formulas**• In the previous sections, we used: • The Fundamental Identities • Sin²x + Cos²x = 1 • Sum & Difference Formulas • Cos (u – v) = Cos u Cos v + Sin u Sin v Now we will use double angle and half angle formulas**Double-Angle Formulas**• Double-angle formulas are the formulas used most often:**Double-Angle Formulas**• Use the following triangle to find the following: Sin 2θ 2 Cos 2θ θ Tan 2θ 5**Double-Angle Formulas**• Use the following triangle to find the following: Sin 2θ = 2Sin θ Cos θ 2 θ 5**Double-Angle Formulas**Cos 2θ = 2Cos² θ - 1 2 θ 5**Double-Angle Formulas**Tan 2θ 2 θ 5**Double-Angle Formulas**• Use the following triangle to find the following: Csc 2θ 1 Sec 2θ θ 4 Cot 2θ**Double-Angle Formulas**• General guidelines to follow when the double-angle formulas to solve equations: • Apply the appropriate double-angle formula • Look to factor • Solve the equation using the different strategies involved in solving equations**Double-Angle Formulas**• Solve the following equation in the interval [0, 2π) Sin 2x – Cos x = 0 1. Apply the double-angle formula 2 Sin x Cos x – Cos x = 0 2. Look to factor Cos x (2 Sin x – 1) = 0**Double-Angle Formulas**Cos x (2 Sin x – 1) = 0 3. Solve the equation Cos x = 0 2 Sin x - 1= 0 Sin x = ½ x x**Double-Angle Formulas**• Solve the following equation in the interval [0, 2π) 2 Cos x + Sin 2x = 0 2 Cos x + 2 Sin x Cos x = 0 2 Cos x (1+ Sin x) = 0 2 Cos x = 0 1 + Sin x = 0**Double-Angle Formulas**2 Cos x = 0 1 + Sin x = 0 Cos x = 0 Sin x = -1 x x**Double-Angle Formulas**• Solve the following equations for x in the interval [0, 2π) • Sin 2x Sin x = Cos x • Cos 2x + Sin x = 0 x x**Double-Angle Formulas**Sin 2x Sin x = Cos x 2 Sin x Cos x Sin x = Cos x 2 Sin²x Cos x – Cos x = 0 Cos x (2 Sin²x – 1) = 0 Cos x = 0 2 Sin²x – 1 = 0 Sin²x = ½ x Sin x = ± ½ x =**Double-Angle Formulas**Cos 2x + Sin x = 0 1 – 2Sin² x + Sin x = 0 2Sin² x - Sin x - 1= 0 (2 Sin x + 1) (Sin x – 1) = 0 2 Sin x + 1 = 0 Sin x – 1 = 0 Sin x = ½ Sin x = 1 x = x**Double-Angle and Half-Angle Formulas**Section 5.5**Double-Angle Formulas**• Evaluating Functions Involving Double Angles Use the given information to find the following: Sin 2x Cos 2x Tan 2x**Double-Angle Formulas**13 12 x -5 Sin 2x = 2Sin x Cos x**Double-Angle Formulas**13 12 x -5 Cos 2x = 2Cos² x - 1**Double-Angle Formulas**13 12 Tan 2x x -5**Double-Angle Formulas**• Evaluating Functions Involving Double Angles Use the given information to find the following: Sin 2x Cos 2x Tan 2x**Double-Angle Formulas**8 x -15 17 Sin 2x = 2Sin x Cos x**Double-Angle Formulas**8 x -15 17 Cos 2x = 2Cos² x - 1**Double-Angle Formulas**8 x -15 Tan 2x 17**The next (and final) set of formulas we have are called**half-angle formulas. The sign of Sin and Cos depend on what quadrant u/2 is in**Half-Angle Formulas**• Use the following triangle to find the six trig functions of θ/2 25 7 θ**Half-Angle Formulas**25 7 θ 24**Half-Angle Formulas**25 7 θ 24**Half-Angle Formulas**25 7 θ 24**Half-Angle Formulas**Find the exact value of the Cos 165º. 165º is half of what angle? Cos 165º =**Half-Angle Formulas**Find the exact value of the Sin 105º. 105º is half of what angle? Sin 105º =**Half-Angle Formulas**Find the exact value of the Tan 15º. 15º is half of what angle? Tan 15º =**Double-Angle and Half-Angle Formulas**Section 5.5**Half-Angle Formulas**13 12 x -5**Half-Angle Formulas**13 12 x -5**Half-Angle Formulas**13 12 x -5**Half-Angle Formulas**13 12 x -5**Half-Angle Formulas**5 3 x 4**Half-Angle Formulas**5 3 x 4**Half-Angle Formulas**5 3 x 4**Half-Angle Formulas**5 3 x 4**Half-Angle Formulas**• Solving Equations using the half-angle formulas: • Apply the appropriate formula • Use the various methods we have learned to solve equations • Factor • Combine Like Terms • Isolate the Trig Function • Solve the Equation for an Angle(s)**Half-Angle Formulas**• Solve the following equation for x in the interval [0, 2π)**Half-Angle Formulas**• Solve the following equation for x in the interval [0, 2π)**Half-Angle Formulas**• Solve the following equation for x in the interval [0, 2π)

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