Section 5.5

 In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u Cos v + Sin u Sin v Now we will use double angle and half angle formulas

 Double-angle formulas are the formulas used most often:

 Use the following triangle to find the following: 2 5 Sin 2 θ Cos 2 θ Tan 2 θ θ

 Use the following triangle to find the following: 2 5 Sin 2 θ = 2Sin θ Cos θ θ

2 5 Cos 2 θ = 2Cos² θ - 1 θ

2 5 Tan 2 θ θ

 Use the following triangle to find the following: 1 4 Csc 2 θ Sec 2 θ Cot 2 θ θ

 General guidelines to follow when the double-angle formulas to solve equations: 1) Apply the appropriate double-angle formula 2) Look to factor 3) Solve the equation using the different strategies involved in solving equations

 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

3. Solve the equation Cos x = 02 Sin x - 1= 0 Sin x = ½ x x

 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 = 01 + Sin x = 0

2 Cos x = 01 + Sin x = 0 Cos x = 0 Sin x = -1 x x

 Solve the following equations for x in the interval [0, 2π) a) Sin 2x Sin x = Cos x b) Cos 2x + Sin x = 0 x x

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 = 02 Sin²x – 1 = 0 Sin²x = ½ Sin x = ± ½ x = x

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 = 0Sin x – 1 = 0 Sin x = ½Sin x = 1 x x =

Section 5.5

 Evaluating Functions Involving Double Angles Use the given information to find the following: Sin 2xCos 2x Tan 2x

12 13 x -5 Sin 2x =2Sin x Cos x

12 13 x -5 Cos 2x =2Cos² x - 1

12 13 x -5 Tan 2x

 Evaluating Functions Involving Double Angles Use the given information to find the following: Sin 2xCos 2x Tan 2x

x 8 Sin 2x =2Sin x Cos x

Cos 2x =2Cos² x x 8

x 8 Tan 2x

 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

 Use the following triangle to find the six trig functions of θ/ θ

7 θ 24

25 7 θ 24

25 7 θ 24

Find the exact value of the Cos 165 º. 165 º is half of what angle? Cos 165 º =

Find the exact value of the Sin 105 º. 105 º is half of what angle? Sin 105 º =

Find the exact value of the Tan 15 º. 15 º is half of what angle? Tan 15 º =

Section 5.5

13 12 x -5

13 12 x -5

13 12 x -5

13 12 x -5

4 3 x 5

4 3 x 5

4 3 x 5

4 3 x 5

 Solving Equations using the half-angle formulas: 1) Apply the appropriate formula 2) Use the various methods we have learned to solve equations 1)Factor 2)Combine Like Terms 3)Isolate the Trig Function 4)Solve the Equation for an Angle(s)

 Solve the following equation for x in the interval [0, 2π)

Because we squared both sides, check your answers!