Solving Trigonometric Equations Unit 5 Solving Trigonometric Equations

Warm up Rewrite as a single trig function and angle. sin 𝜋/4 cos 𝜋/3 − cos 𝜋/4 sin 𝜋/3 −𝑠𝑖𝑛 𝜋 12

You Try/Warm up! Find tan(a+b) if sin a=1/3 and cos b=-24/25 a and b are both in QII Find the exact value of sin 5𝜋 12 And now for more trig formulas... −24−7 8 24 8 −7 sin 𝜋 4 + 𝜋 6 = 6 + 2 4

Question??? What’s the difference between sin(2x)=-√3/2 and sin (2x)?? Well… we know that sin(2x)=-√3/2 is on the unit circle and we know how to solve this problem Sin(2x) doesn’t give us enough information to know if it’s on the unit circle or not. So we need a new technique. The new technique is call the double angle formulas.

Double Angle Formulas

Double Angle Formulas Example 1 Find sin(2x), cos(2x), and tan(2x) given cos x=-(24/25) where 𝜋<𝑥< 3𝜋 2 sin 2𝑥 =2 sin 𝑥 cos 𝑥 =2 −7 25 −24 25 = 336 625 cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥− 𝑠𝑖𝑛 2 𝑥 = −24 25 2 − −7 25 2 = 576−49 625 = 527 625 tan 2𝑥 = 2 tan 𝑥 1− 𝑡𝑎𝑛 2 𝑥 = 2 7 24 1− 7 24 2 = 14 24 527 576 = 14 24 576 527 = 336 527 24 7 25 sin 𝑥= −7 25 cos 𝑥= −24 25 tan 𝑥=+ 7 24

Double Angle Formula Example 2 Substitution Simplify: 1−𝑐𝑜𝑠 2𝑥 𝑠𝑖𝑛 2𝑥 1− 1−2 𝑠𝑖𝑛 2 𝑥 2 sin 𝑥 cos 𝑥 2 𝑠𝑖𝑛 2 𝑥 2 sin 𝑥 cos 𝑥 sin 𝑥 cos 𝑥 tan 𝑥

Half Angle Formulas Note the sign of sin and cos depend on the quadrant in which a/2 lies

Half Angle Formula Example 1 𝑥 2 = 2.7 2 =1.4 Q1 13 5 12 Sin x= 𝟓 𝟏𝟑 and is in QII Find sin( 𝒙 𝟐 ), cos( 𝒙 𝟐 ), tan( 𝒙 𝟐 ) sin 𝑥 2 =+ 1− −12 13 2 = 25 13 1 2 =5 1 26 =5 26 26 cos 𝑥 2 =+ 1+ −12 13 2 = 1 13 1 2 = 26 26 tan 𝑥 2 = 𝑠𝑖𝑛 𝑥 2 𝑐𝑜𝑠 𝑥 2 5 26 26 26 26 =5

Half Angle Formula You Try Find the exact value of cos 3𝜋 8 Notice that 3𝜋 8 is half of 3𝜋 4 3𝜋 8 is in what quadrant? a/2 is in what quadrant? = 1+ cos 3𝜋 4 2 = 1+ − 2 2 2 = 2− 2 2 2 = 2− 2 4 = 2− 2 2

Exit Ticket How do you know when to use multiple angles instead of double or half angle formulas? WebAssign #2 Due Friday