1. Section 3.8 Solving Trigonometric Equations and HC
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2. I) Review: Solving
“Solving” means finding a value for the variable, so that both sides of an equation will be equal
There are several ways to solve a Trigonometric equation:
Algebraically: (Ch5.2)
Isolate the variable using BEDMAS
Use inverse trig. functions to find the solution(s)
There are usually two solutions within one cycle (CAST)
Use the reference angles to find the solution in standard position
Graphically:
Make Y1 equal to one side of the function
Make Y2 equal to the other side
Then find the points of intersection(s)
Points where the graph intersect  the x-coordinates are the solutions
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3. Ex: Solve the Equation for 0 ≤ θ ≤ 3π
Make the left side of the equation y1
Make the right side of the equation y2
Solve the system by finding the points of intersection x y p 2p 3p
-14
-13
-12
-11
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
Press:
2nd
TRACE
#5: Intersect
1st Curve?
ENTER
2nd Curve?
ENTER
Guess?
ENTER
Repeat this process until you find all the points of intersection
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4. Practice: Solve the Equation for 0 ≤ x ≤ 2πy
-2p -p p 2p 1 2 3 4 5 6 7
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5. II) Finding the General solution:
When finding the general solution, only one pair of solutions within a cycle is needed
Other solutions can be founded by adding/subtracting multiples of the period (Co-terminal angles)
Graphically, the horizontal distance between 2 solutions in separate cycles is equal to the period
If you want to find solutions within other cycles, then just add/subtract the period x y p 2p 3p
-14
-13
-12
-11
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
The two general solutions will be:
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6. III) Solving Trigonometric Functions with Horizontal Compressions:
Ex: Find the value of θ, for 0 ≤ θ ≤ 360°
The sine function is compressed horizontally by a factor of 3
The number of solutions will increase from 2 to 6 solutions between 0° and 360° x y p 2p -1 1
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7. Ex: Given the following system, how many solutions will there be between 0 ≤ θ ≤ 360° ,where “B” is a constant
The constant “B” will indicate the horizontal factor of expansion/compression
If B=2, then the graph will compress horizontally by a factor of ½. There will be 2 cycles between 0 & 360° x y
0.5 1
1.5 2
2.5 3
3.5 4
4.5 5
5.5 6 -1
If B=3, then the graph will compress horizontally by a factor of 1/3. There will be 3 cycles between 0 & 360° x y
0.5 1
1.5 2
2.5 3
3.5 4
4.5 5
5.5 6 -1
The number of solutions will be 2 x B
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8. iv) Solving Trigonometric Equations Algebraically
Isolate the variable using BEDMAS
Use inverse trig. functions to find the solution(s)
There are usually two solutions within one cycle (CAST)
Use the reference angles to find the solution in standard position
When there is a horizontal compression, add the period to find other sets of solutions
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9. v) Solving Equations with Horizontal Compressions
Ex: Solve for “x”, 0 ≤ x ≤ 2π
Make the substitution: 3x = θ
Isolate the trigonometric Function
Solve for θ in Quadrant 2
To Solve for “x”, substitute 3x back into θ
Add the period to find solutions in the next cycle
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10. Practice: Solve for “x”, 0 ≤ x ≤ 2π
Make the substitution: 2x = θ
Isolate the trigonometric Function
Solve for θ in Quadrant 4
To Solve for “x”, substitute 2x back into θ
Add the period to find solutions in the next cycle
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11. VI) Solving Trigonometric equations Using Algebra
Ex: Solve for “x”, 0 ≤ x ≤ 2π
Factor
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12. Ex: Solve for “x”, 0 ≤ x ≤ 2π Factor
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13. VII) Solving Trigonometric equations Involving Substitution
Ex: Solve for “x”, 0 ≤ x ≤ 2π
Make the substitution: sin x = T
Substitute T back into sin x
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14. Practice: Solve for “x”, 0 ≤ x ≤ 2π
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15. Viii) Using Double Angle Identities to Solve Equations
Ex: Solve the following equation for 0 ≤ θ ≤ 360°
Note: You are “Solving for θ”, not proving an identity!!
Use the Double-Angle Identity
Factor the equation
Solve for θ from each equation.
Remember to use CAST and there
are TWO cycles between 0 and 360°
Add the period (180°) to find other solutions
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16. Practice: Solve for 0 ≤ θ ≤ 360°
Use the Double-Angle Identity
Factor: Difference of Square
Solve for θ from each equation.
Remember to use CAST
Note: There is only ONE cycle
Between 0 and 360°
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17. Challenge: Solve for “x”, 0 ≤ x ≤ 2π
Make the substitution:
tan(2x) = T
Add the period to find solutions in the next cycle
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18. Check: x y
0.5p p
1.5p 2p
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19. Homework
Assignment 3.8