Trigonometric Equations and Identities

1
Chapter 19

• Trigonometric Equations and Identities

2
Table of Contents

• Basic Trigonometric Identities Slide 3
• Types of Expressions Slide 7
• Solving Trigonometric Equations Slide 11
• Solving Second Degree (Quadratic) Trigonometric
  Equations Slide 16

3
Basic Trigonometric Identities

• Notes

4
Basic Trigonometric Identities
5
Basic Trigonometric Identities
6
Basic Trigonometric Identities
7
Types of Expressions

• Notes

8
Types of Expressions

• An equation has a unique solution. It is an
  expression that is only true for certain
  replacement values.
• An identity is true for all replacement values.

9
Types of Expressions

• Equation
• 2x 1 7
• x 3
• Only true when x 3.

• Identity
• 3x 4x 7x
• This is always true no matter what values are
  substituted for x.

10
Types of Expressions

• Example Rewrite each expression in terms of sin
  ?, cos ?, or a constant.
• 1)
• 2)
• 3)

11
Solving Trigonometric Equations

• Notes

12
Solving Trigonometric Equations

• Example Solve for x.
• 0 x 360
• 2cosx 1 0
• 2cosx 1
• cosx ½
• x 60

13
Solving Trigonometric Equations

• Remember
• The answer to the problem is your reference
  angle.
• In the first quadrant, the answer equals the
  reference angle.
• In the second quadrant, the answer equals 180
  reference angle.
• In the third quadrant, the answer equals 180
  reference angle.
• In the fourth quadrant, the answer equals 360
  reference angle.

14
Solving Trigonometric Equations

• Example Solve for x to the nearest ten
  minutes.
• 0 x 360
• 2tanx 3 2
• 2tanx -1
• tanx -½
• x 2634

15
Solving Trigonometric Equations

• When using the calculator, do not enter the
  negative sign when pressing 2ND TAN.
• The negative sign is used to determine the
  quadrant that the angle lies in (Unit Circle).
• Therefore, the answer 2634 is the reference
  angle.
• The angle lies in the second and fourth
  quadrants, therefore, the angles are 15330 and
  33330.

16
Solving Second Degree (Quadratic) Trigonometric
Equations

• Notes

17
Solving Second Degree Equations

• Example Solve. 0 x 360
• tan2x 3tanx 4 0
• let x tan x
• x2 3x 4 0
• (x 4)(x 1) 0
• x 4 0 x 1 0
• x 4 x -1
• tan x 4 tan x -1
• x 76 x 135
• x 256 x 315

18
Solving Second Degree Equations

• Example Solve. 0 x 360
• 3cos2x 5cosx 4
• let x cos x
• 3x2 5x 4 0

19
Solving Second Degree Equations

• x 2.26 x -0.59
• cos x 2.26 cos x -0.59
• x Ø x 126
• x 234

20
Solving Second Degree Equations

• Example Solve. 0 x 360
• 2cos2x cosx
• let x cos x
• 2x2 x
• 2x2 x 0
• x(2x 1) 0
• x 0 2x 1 0
• x .5
• cos x 0 cos x .5
• x 90 x 270
• x 270

21
Solving Second Degree Equations

• Example Solve. 0 x 360
• 3sinx 4 1/sinx
• let x sin x
• 3x 4 1/x
• x(3x 4) x(1/x)
• 3x2 4x 1
• (3x 1)(x 1) 0

22
Solving Second Degree Equations

• 3x - 1 0 x 1 0
• x .333 x -1
• sin x .333 sin x -1
• x 19 x 270
• x 161

23
Solving Second Degree Equations

• Example Solve. 0 x 360
• 2cos2x sinx 1
• Use the identity cos2x 1 sin2x
• 2 (1 sin2x) sinx 1
• 2 2sin2x sinx 1
• 2sin2x sinx 1 0
• 2sin2x sinx - 1 0

24
Solving Second Degree Equations

• 2sin2x sinx - 1 0
• let x sin x
• 2x2 x - 1 0
• (2x - 1)(x 1) 0
• 2x - 1 0 x 1 0
• x ½ x -1
• sin x ½ sin x -1
• x 30 x 270
• x 150

25
Chapter 19

• End of Chapter