1. Trigonometric Equations
In quadratic form, using identities or linear in sine and cosine

2. Solving a Trig Equation in Quadratic Form
Solve the equation:
2sin2 θ – 3 sin θ + 1 = 0, 0 ≤ θ ≤ 2p
Let sin θ equal some variable
sin θ = a
Factor this equation
(2a – 1) (a – 1) = 0
Therefore a = ½ a = 1

3. Solving a Trig Equation in Quadratic Form
Now substitute sin θ back in for a
sin θ = ½ sin θ = 1
Now do the inverse sin to find what θ equals
θ = sin-1 (½) θ = sin-1 1
θ = p/6 and 5p/6 θ = p/2

4. Solving a Trig Equation in Quadratic Form
Solve the equation:
(tan θ – 1)(sec θ – 1) = 0
tan θ – 1 = 0 sec θ – 1 = 0
tan θ = 1 sec θ = 1
θ = tan θ = sec-1 1
θ = p/4 and 5p/4 θ = 0

5. Solving a Trig Equation Using Identities
In order to solve trig equations, we want to have a single trig word in the equation. We can use trig identities to accomplish this goal.
Solve the equation
3 cos θ + 3 = 2 sin2 θ
Use the pythagorean identities to change sin2 θ to cos θ

6. Solving a Trig Equation Using Identities
sin2 = 1 – cos2 θ
Substituting into the equation
3 cos θ + 3 = 2(1 – cos2 θ)
To solve a quadratic equation it must be equal to 0
2cos2 θ + 3 cos θ + 1 = 0
Let cos θ = b

7. Solving a Trig Equation using Identities
2b2 + 3b + 1 = 0
(2b + 1) (b + 1) = 0
(2b + 1) = 0 b + 1 = 0
b = -½ b = -1
cos θ = -½ cos θ = -1
θ = 2p/3, 4p/3 θ = p

8. Solving a Trig Equation Using Identities
cos2 θ – sin2 θ + sin θ = 0
1 – sin2 θ – sin2 θ + sin θ = 0
-2sin2 θ + sin θ + 1 = 0
2 sin2 θ – sin θ – 1 = 0
Let c = sin θ
2c2 – c – 1 = 0
(2c + 1) (c – 1) = 0

9. Solving a Trig Equation Using Identities
(2c + 1) = 0 c – 1 = 0
c = -½ c = 1
sin θ = -½ sin θ = 1
θ = p/3 + p q=2p-p/3 θ = p/2
θ = 4p/3, q = 7p/3

10. Solving a Trig Equation Using Identities
Solve the equation
sin (2θ) sin θ = cos θ
Substitute in the formula for sin 2θ
(2sin θ cos θ)sin θ=cos θ
2sin2 θ cos θ – cos θ = 0
cos θ(2sin2 – 1) = 0
cos θ = 0 2sin2 θ=1

11. Solving a Trig Equation Using Identities
cos θ = 0
θ = 0, p θ = p/4, 3p/4, 5p/4, 7p/4

12. Solving a Trig Equation Using Identities
sin θ cos θ = -½
This looks very much like the sin double angle formula. The only thing missing is the two in front of it.
So multiply both sides by 2
2 sin θ cos θ = -1
sin 2θ = -1
2 θ = sin-1 -1

13. Solving a Trig Equation Using Identities
2θ = 3p/2
θ = 3p/4 2θ = 3p/2 + 2p
2q = 7p/2
q = 7p/4

14. Solving a Trig Equation Linear in sin θ and cos θ
sin θ + cos θ = 1
There is nothing I can substitute in for in this problem. The best way to solve this equation is to force a pythagorean identity by squaring both sides.
(sin θ + cos θ)2 = 12

15. Solving a Trig Equation Linear in sin θ and cos θ
sin2 θ + 2sin θ cos θ + cos2 θ = 1
2sin θ cos θ + 1 = 1
2sin θ cos θ = 0
sin 2θ = 0
2θ = 0 2θ = p
θ = 0 θ = p/2
θ = p θ = 3p/2

16. Solving a Trig Equation Linear in sin θ and cos θ
Since we squared both sides, these answers may not all be correct (when you square a negative number it becomes positive).
In the original equation, there were no terms that were squared

17. Solving a Trig Equation Linear in sin θ and cos θ
Check:
Does sin 0 + cos 0 = 1?
Does sin p/2 + cos p/2 = 1?
Does sin p + cos p = 1?
Does sin 3p/2 + cos 3p/2 = 1?

18. Solving a Trig Equation Linear in sin θ and cos θ
sec θ = tan θ + cot θ
sec2 θ = (tan θ + cot θ)2
sec2 θ = tan2 θ + 2 tan θ cot θ + cot2 θ
sec2 θ = tan2 θ cot2 θ
sec2 θ – tan2 θ = 2 + cot2 θ
1 = 2 + cot2 θ
-1 = cot2 θ

19. Solving a Trig Equation Linear in sin θ and cos θ
q is undefined (can’t take the square root of a negative number).

20. Solving Trig Equations Using a Graphing Utility
Solve 5 sin x + x = 3. Express the solution(s) rounded to two decimal places.
Put 5 sin x + x on y1
Put 3 on y2
Graph using the window 0 ≤ θ ≤2p
Find the intersection point(s)

21. Word Problems
Page 519 problem 58