1. Solving Trigonometric Equations
Unit 5
Solving Trigonometric Equations

2. 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

3. 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
sin 𝜋 4 + 𝜋 6 =

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.

5. Double Angle Formulas

6. 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 − − =
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥− 𝑠𝑖𝑛 2 𝑥
= − − − = 576− =
tan 2𝑥 = 2 tan 𝑥 1− 𝑡𝑎𝑛 2 𝑥 = − = = = 24 7 25
sin 𝑥= −7 25
cos 𝑥= −24 25
tan 𝑥=+ 7 24

7. 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

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

9. Half Angle Formula Example 1
𝑥 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− − = = =
cos 𝑥 2 = − = =
tan 𝑥 2 = 𝑠𝑖𝑛 𝑥 2 𝑐𝑜𝑠 𝑥 =5

10. 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
= − = 2− = 2− = 2−

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