Ch 5.5.

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Presentation transcript:

Ch 5.5

HW questions?

4 more categories of trig identities Multiple angles EX: sin(ku) cos(ku) Squares of trig functions EX: 𝑠𝑖𝑛 2 𝑥 𝑐𝑜𝑠 2 𝑥 Half angles EX: sin( 𝑥 2 ) cos( 𝑥 2 ) Products of trig functions EX: cos(u)sin(v)

EX 1: Evaluating functions involving double angles Given cos(Ɵ) = 5 13 and 3П 2 < Ɵ < 2П Find sin(2 Ɵ) 2) Cos(2 Ɵ ) 3)Tan(2 Ɵ )

EX 2:Solving a Multiple-Angle Equation Solve 2 cos(x) + sin(2x) = 0 aka what angles make this equation true? 2cos(x) + 2sin(x)cos(x) = 0 Use Double-Angle formula 2cos(x)[1+ sin(x)] = 0 Factor 2cos(x) = 0 and 1 + sin(x) = 0 Set each to 0 and solve cos(x) = 0 sin(x) = -1 x = П 2 and 3П 2 and x = 3П 2 on the interval [0, 2П) General solution: x = П 2 + Пn

EX 3: Simplify y = 4 cos²(x) – 2 using the double angle formulas EX 3: Simplify y = 4 cos²(x) – 2 using the double angle formulas. Then graph.

EX 5: Using the Half Angle Formulas to find exact values of trig functions Find the exact value of sin(105ᴼ) Notice that 105ᴼ is half of 210ᴼ, which is one of our special angles. Sin is positive in quadrant II so we have: Sin(105ᴼ) = 1 −cos⁡(210ᴼ) 2 = 1 −(− 3 /2) 2 = 2+ 3 2 If an angle is half of one of our “special angles,” we can use the half angle formulas to find exact values.