8.3: Algebraic Solutions of
Trigonometric Equations
© 2008 Roy L. Gover(www.mrgover.com)
Learning Goals:
•Solve Trigonome...
Try This
Using your knowledge of
Algebra and factoring, solve
the equation:
2
0x x+ =
0, 1x x= = −
Check your answer.
Example
Solve the equation and
check the solution:
2
sin sin 0x x+ =
Important Idea
Always check your
solution(s) in the original
equation.
Try This
Using your knowledge of
Algebra, solve the equation:
2 3 9x + =
3x =
Check your answer.
Example
Solve the equation and
check the solution:
2cos 1 3θ + =
Important Idea
Similar methods are
used to solve algebraic
and trigonometric
equations.
Example
Solve for all values:
…there are an infinite
number of solutions.
2cos 3x =
Definition
Principal Values in Degrees:
& tan x: 90 90x− ° ≤ ≤ °
cos x: 0 180x° ≤ ≤ °
Principal values
insure unique
sin x
Definition
Principal Values in
Radians:
sin x & tan x:
2 2
x
π π
− ≤ ≤ +
cos x: 0 x π≤ ≤
Principal values
insure unique
Try This
Solve for the principal value:
2cos 1 0x − =
60x = ° or
3
π
Try This
Solve for the principal
value in degrees:
30x = − °
What is the
answer in
rads? 6
x
π
= −
2sin 1 0x + =
Try This
Solve for the principal
value in radians:
3sin 1x =
.340x =
Important Idea
When you
cannot get
exact values
for solutions,
you must use
the inverse trig
functions on
your calculator....
Try This
Solve for the principal
value in degrees:
8cos 5 0θ − =
51.318x = °
Example
Solve for all values in
degrees:
8cos 5 0θ − =
Try This
Solve: 2sin 5 0x − =
???????
Important Idea
The value of sin x and cos x
can never be greater than 1
or less than -1
Range:
sin x : [-1,1]
cos x : [-1,...
Try This
Using your knowledge of
Algebra and factoring, solve
the equation:
2
2 0x x− − =
1, 2x x= − =
Check your answer.
Example
Solve the equation and
check the solution:
2
sin sin 2 0x x− − =
Try This
Solve the equation and
check the solution:
2
cos cos 2 0x x+ − =
0x =
Are there other solutions?
Example
Find all solutions of
2
3sin sin 2 0x x− − =
Try This
Solve for the principal
values in radians:
2
2sin sin 1 0x x+ − =
6
x
π
= or
2
π
−
Why is not an answer?
3
2
π
Example
Find all solutions in
radians:
2
tan cos tanx x x=
Try This
Find the principal value
solutions in degrees:
2
cos sin cosx x x=
90x = ° 90x = − °or
Example
Solve:
2
10cos 3sin 9 0x x− − + =
Example
Solve:
2
sec 5tan 2x x+ = −
Important Idea
2 2
sin cos 1x x+ =
2 2
sin 1 cosx x= −
2 2
cos 1 sinx x= −
Trigonometric Identities:
2 2
tan 1 secx x+ =
2...
Important Idea
2
4
2
b b ac
x
a
− ± −
=
2
0ax bx c+ + =for
The quadratic formula:
Try This
Solve for principal values
in radians:
2
sec 2tan 3x x− =
x=1.220 and x= -.632
Important Idea
When solving trig identities,
try one or more of the
following:
•Factor if possible
•Use quadratic formula ...
Lesson Close
Trignometric equations are
used in science and
engineering.
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  • P542, ex 7
  • Ex8 p 543
  • P544 ex9: factors (2sinx-1)(5sinx+1) after using identity
  • Ex 10, p544Use identity and quadratic formula
  • Hprec8 3

    1. 1. 8.3: Algebraic Solutions of Trigonometric Equations © 2008 Roy L. Gover(www.mrgover.com) Learning Goals: •Solve Trigonometric equations using Algebraic methods.
    2. 2. Try This Using your knowledge of Algebra and factoring, solve the equation: 2 0x x+ = 0, 1x x= = − Check your answer.
    3. 3. Example Solve the equation and check the solution: 2 sin sin 0x x+ =
    4. 4. Important Idea Always check your solution(s) in the original equation.
    5. 5. Try This Using your knowledge of Algebra, solve the equation: 2 3 9x + = 3x = Check your answer.
    6. 6. Example Solve the equation and check the solution: 2cos 1 3θ + =
    7. 7. Important Idea Similar methods are used to solve algebraic and trigonometric equations.
    8. 8. Example Solve for all values: …there are an infinite number of solutions. 2cos 3x =
    9. 9. Definition Principal Values in Degrees: & tan x: 90 90x− ° ≤ ≤ ° cos x: 0 180x° ≤ ≤ ° Principal values insure unique sin x
    10. 10. Definition Principal Values in Radians: sin x & tan x: 2 2 x π π − ≤ ≤ + cos x: 0 x π≤ ≤ Principal values insure unique
    11. 11. Try This Solve for the principal value: 2cos 1 0x − = 60x = ° or 3 π
    12. 12. Try This Solve for the principal value in degrees: 30x = − ° What is the answer in rads? 6 x π = − 2sin 1 0x + =
    13. 13. Try This Solve for the principal value in radians: 3sin 1x = .340x =
    14. 14. Important Idea When you cannot get exact values for solutions, you must use the inverse trig functions on your calculator. 1 sin x− 1 cos x− 1 tan x− • • •
    15. 15. Try This Solve for the principal value in degrees: 8cos 5 0θ − = 51.318x = °
    16. 16. Example Solve for all values in degrees: 8cos 5 0θ − =
    17. 17. Try This Solve: 2sin 5 0x − = ???????
    18. 18. Important Idea The value of sin x and cos x can never be greater than 1 or less than -1 Range: sin x : [-1,1] cos x : [-1,1]
    19. 19. Try This Using your knowledge of Algebra and factoring, solve the equation: 2 2 0x x− − = 1, 2x x= − = Check your answer.
    20. 20. Example Solve the equation and check the solution: 2 sin sin 2 0x x− − =
    21. 21. Try This Solve the equation and check the solution: 2 cos cos 2 0x x+ − = 0x = Are there other solutions?
    22. 22. Example Find all solutions of 2 3sin sin 2 0x x− − =
    23. 23. Try This Solve for the principal values in radians: 2 2sin sin 1 0x x+ − = 6 x π = or 2 π − Why is not an answer? 3 2 π
    24. 24. Example Find all solutions in radians: 2 tan cos tanx x x=
    25. 25. Try This Find the principal value solutions in degrees: 2 cos sin cosx x x= 90x = ° 90x = − °or
    26. 26. Example Solve: 2 10cos 3sin 9 0x x− − + =
    27. 27. Example Solve: 2 sec 5tan 2x x+ = −
    28. 28. Important Idea 2 2 sin cos 1x x+ = 2 2 sin 1 cosx x= − 2 2 cos 1 sinx x= − Trigonometric Identities: 2 2 tan 1 secx x+ = 2 2 1 cot cscx x+ = See section 6.5 of your text for additional trig
    29. 29. Important Idea 2 4 2 b b ac x a − ± − = 2 0ax bx c+ + =for The quadratic formula:
    30. 30. Try This Solve for principal values in radians: 2 sec 2tan 3x x− = x=1.220 and x= -.632
    31. 31. Important Idea When solving trig identities, try one or more of the following: •Factor if possible •Use quadratic formula if you can’t factor •Use identities to write the equation in terms of only one trig function.
    32. 32. Lesson Close Trignometric equations are used in science and engineering.
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