1. Solving Trigonometric Equations

2. Solving Trig Equations
When solving trig equations, you will need to get the trig function isolated (by itself).
Ex: 2sin x = Divide both sides by 2
sin x = Use unit circle or inverse
function on calculator to find angle
We will limit our solutions to 0, 2π , and all answers must be in RADIANS (p form)
30 𝑜 𝑎𝑛𝑑 150 𝑜 = 𝜋 6 𝑎𝑛𝑑 5𝜋 6 S A T C

3. S A T C Answers: 𝜋 4 𝑎𝑛𝑑 3𝜋 4 Ex: sin x - 2 = - sin x now add sinx to
both sides.
2sin x = add 𝟐 to both sides.
2sin x = divide by 2 (keep as fraction)
sin x = find angle with sin
Look only at the interval from 0, 2𝜋 to find angles
Answers: 𝜋 4 𝑎𝑛𝑑 3𝜋 4 S A T C

4. S A T C Ex: 4 𝒔𝒊𝒏 𝟐 𝒙 – 3 = 0 now add 3 both sides.
4 𝑠𝑖𝑛 2 𝑥 = divide by 4.
𝑠𝑖𝑛 2 𝑥= take the √ of both sides.
sin 𝑥 =± use unit circle or calculator
Which angles have a sine value that is ± ?
Answers: 𝜋 3 , 2𝜋 3 , 4𝜋 3 , 5𝜋 3 S A T C

5. sin 𝑥 =0 sin 𝑥 −2=0 solve both sin 𝑥 =2
How do you solve a problem involving factoring?
(Set equation equal to zero and then factor)
Ex: 𝒔𝒊𝒏 𝟐 𝒙=𝟐 𝒔𝒊𝒏 𝒙 subtract 2 sin x
𝑠𝑖𝑛 2 𝑥 −2 sin 𝑥 =0 factor out sin x
sin 𝑥 sin 𝑥 −2 = set each factor = 0
sin 𝑥 = sin 𝑥 −2= solve both
sin 𝑥 =2
Look only at the interval from 0, 2𝜋
Answers: 𝟎 , π 𝒐𝒏𝒍𝒚 since sin x ≠ 2 ever

6. How do you solve problems of a quadratic type?
(Set equation equal to zero and then factor OR…
quadratic formula)
Ex: 𝟐𝒔𝒊𝒏 𝟐 𝒙−𝟑𝒔𝒊𝒏𝒙+𝟏=𝟎 factor
2 sin 𝑥 −1)( sin 𝑥 −1 =0 set each factor = 0
2 sin 𝑥 −1 = 𝑎𝑛𝑑 sin 𝑥 −1 =0
2 sin 𝑥 = sin 𝑥 =1
sin 𝑥 = 1 2
Look only at the interval from 𝟎, 𝟐𝝅 to find angles
For sin ½  𝝅 𝟔 , 𝟓𝝅 𝟔 For sin 1  𝝅 𝟐

7. What if there is more than one trig function in the equation?
(Use identities to change into one trig term, then factor and solve)
Ex: 𝟑𝒔𝒆𝒄 𝟐 𝒙− 𝟐𝒕𝒂𝒏 𝟐 𝒙−𝟒 =𝟎 rewrite 𝒔𝒆𝒄 𝟐 𝒙 using an identity
3 (1+𝑡𝑎𝑛 2 𝑥) −2 𝑡𝑎𝑛 2 𝑥−4= distribute
3+3 𝑡𝑎𝑛 2 𝑥−2 𝑡𝑎𝑛 2 𝑥 −4= simplify
𝑡𝑎𝑛 2 𝑥−1= add 1 to both sides
𝑡𝑎𝑛 2 𝑥= take the square root
𝐭𝐚𝐧𝐱= ± 𝟏 what angle has a tangent value of ±𝟏?
In the interval [0, 2𝝅), the answers are:
𝝅 𝟒 , 𝟑𝝅 𝟒 , 𝟓𝝅 𝟒 , 𝟕𝝅 𝟒

8. Squaring and converting to quadratic type
Ex: 𝑠𝑖𝑛𝑥+1= cos 𝑥 it is unclear how to rewrite
this equation as one trig function Instead, it may be easier to square
both sides. (Be careful when squaring)
(𝑠𝑖𝑛𝑥+1) 2 = 𝑐𝑜𝑠 2 𝑥 now cos2 can be replaced
sin 𝑥+1 𝑠𝑖𝑛𝑥+1 =1 − 𝑠𝑖𝑛 2 𝑥
𝑠𝑖𝑛 2 𝑥+2𝑠𝑖𝑛𝑥+1=1 − 𝑠𝑖𝑛 2 𝑥 get equation set = 0
2 𝑠𝑖𝑛 2 𝑥+2 𝑠𝑖𝑛𝑥= Factor by GCF of 2 sin x

9. 2𝑠𝑖𝑛𝑥 𝑠𝑖𝑛𝑥+1 =0 set each factor to 0 and solve 2𝑠𝑖𝑛𝑥=0 𝑎𝑛𝑑 𝑠𝑖𝑛𝑥+1=0
2𝑠𝑖𝑛𝑥= 𝑎𝑛𝑑 𝑠𝑖𝑛𝑥+1=0
𝑠𝑖𝑛𝑥= 𝑠𝑖𝑛𝑥=−1
Look only at the interval from 0, 2π to find angles
For sin 0  𝟎, 𝝅
For sin -1  𝟑𝝅 𝟐