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5.3 Solving Trig equations. Solving Trig Equations. Solve the following equation for x: Sin x = ½ . Solving Trig Equations. In this section, we will be solving various types of trig equations You will need to use all the procedures learned last year in Algebra II

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Solving trig equations
Solving Trig Equations

  • Solve the following equation for x:

    Sin x = ½


Solving trig equations1
Solving Trig Equations

  • In this section, we will be solving various types of trig equations

  • You will need to use all the procedures learned last year in Algebra II

  • All of your answers should be angles.

  • Note the difference between finding all solutions and finding all solutions in the domain [0, 2π)


Solving trig equations2
Solving Trig Equations

  • Guidelines to solving trig equations:

  • Isolate the trig function

  • Find the reference angle

  • Put the reference angle in the proper quadrant(s)

  • Create a formula for all possible answers (if necessary)


Solving trig equations3
Solving Trig Equations

1- 2 Cos x = 0

1) Isolate the trig function

1- 2 Cos x = 0

+ 2 Cos x = + 2 Cos x

1= 2 Cos x

2 2

Cos x = ½


Solving trig equations4
Solving Trig Equations

Cos x = ½

2) Find the reference angle

x =

3) Put the reference angle in the proper quadrant(s)

I =

IV =


Solving trig equations5
Solving Trig Equations

Cos x = ½

4) Create a formula if necessary

x =

x =


Solving trig equations6
Solving Trig Equations

  • Find all solutions to the following equation:

    Sin x + 1 = - Sin x

+ Sin x + Sin x

→ 2 Sin x + 1 = 0

- 1 - 1

→ 2 Sin x = -1

→ Sin x = - ½


Solving trig equations7
Solving Trig Equations

Sin x = - ½

Ref. Angle:

Quad.:

III:

Iv:


Solving trig equations8
Solving Trig Equations

  • Find the solutions in the interval [0, 2π) for the following equation:

    Tan²x – 3 = 0

Tan²x = 3

Tan x =


Solving trig equations9
Solving Trig Equations

Tan x =

Ref. Angle:

Quad.:

I:

III:

IV:

II:

x =


Solving trig equations10
Solving Trig Equations

  • Solve the following equations for all real values of x.

  • Sin x + = - Sin x

  • 3Tan² x – 1 = 0

  • Cot x Cos² x = 2 Cot x


Solving trig equations11
Solving Trig Equations

  • Find all solutions to the following equation:

    Sin x + = - Sin x

2 Sin x = -

x =

Sin x = -

x =


Solving trig equations12
Solving Trig Equations

3Tan² x – 1 = 0

x =

Tan² x =

x =

Tan x =

x =

x =


Solving trig equations13
Solving Trig Equations

Cot x Cos² x = 2 Cot x

Cot x Cos² x – 2 Cot x = 0

Cot x (Cos² x – 2) = 0

Cot x = 0

Cos² x – 2 = 0

Cos x = 0

Cos² x – 2 = 0

x =

Cos x =

No Solution

x =




Solving trig equations15
Solving Trig Equations

  • Find all solutions to the following equation.

    4 Tan²x – 4 = 0

x =

Tan²x = 1

x =

Tan x = ±1

Ref. Angle =


Solving trig equations16
Solving Trig Equations

  • Equations of the Quadratic Type

  • Many trig equations are of the quadratic type:

    • 2Sin²x – Sin x – 1 = 0

    • 2Cos²x + 3Sin x – 3 = 0

  • To solve such equations, factor the quadratic or, if that is not possible, use the quadratic formula


  • Solving trig equations17
    Solving Trig Equations

    • Solve the following on the interval [0, 2π)

      2Cos²x + Cos x – 1 = 0

    2x² + x - 1

    If possible, factor the equation into two binomials.

    (2Cos x – 1) (Cos x + 1) = 0

    Now set each factor equal to zero


    Solving trig equations18
    Solving Trig Equations

    2Cos x – 1 = 0 Cos x + 1 = 0

    Cos x = ½

    Cos x = -1

    Ref. Angle:

    x =

    Quad:

    I, IV

    x =


    Solving trig equations19
    Solving Trig Equations

    • Solve the following on the interval [0, 2π)

      2Sin²x - Sin x – 1 = 0

    (2Sin x + 1) (Sin x - 1) = 0


    Solving trig equations20
    Solving Trig Equations

    2Sin x + 1 = 0 Sin x - 1 = 0

    Sin x = - ½

    Sin x = 1

    Ref. Angle:

    x =

    Quad:

    III, IV

    x =


    Solving trig equations21
    Solving Trig Equations

    • Solve the following on the interval [0, 2π)

      2Cos²x + 3Sin x – 3 = 0

    Convert all expressions to one trig function

    2 (1 – Sin²x) + 3Sin x – 3 = 0

    2 – 2Sin²x + 3Sin x – 3 = 0

    0 = 2Sin²x – 3Sin x + 1


    Solving trig equations22
    Solving Trig Equations

    0 = 2Sin²x – 3Sin x + 1

    0 = (2Sin x – 1) (Sin x – 1)

    2Sin x - 1 = 0 Sin x - 1 = 0

    Sin x = ½

    Sin x = 1

    Ref. Angle:

    x =

    Quad:

    I, II

    x =


    Solving trig equations23
    Solving Trig Equations

    • Solve the following on the interval [0, 2π)

      2Sin²x + 3Cos x – 3 = 0

    Convert all expressions to one trig function

    2 (1 – Cos²x) + 3Cos x – 3 = 0

    2 – 2Cos²x + 3Cos x – 3 = 0

    0 = 2Cos²x – 3Cos x + 1


    Solving trig equations24
    Solving Trig Equations

    0 = 2Cos²x – 3Cos x + 1

    0 = (2Cos x – 1) (Cos x – 1)

    2Cos x - 1 = 0 Cos x - 1 = 0

    Cos x = ½

    Cos x = 1

    Ref. Angle:

    x =

    Quad:

    I, IV

    x =


    Solving trig equations25
    Solving Trig Equations

    • The last type of quadratic equation would be a problem such as:

      Sec x + 1 = Tan x

      What do these two trig functions have in common?

      When you have two trig functions that are related

      through a Pythagorean Identity, you can square

      both sides.

    ( )² ²


    Solving trig equations26
    Solving Trig Equations

    (Sec x + 1)² = Tan²x

    Sec²x + 2Sec x + 1

    = Sec²x - 1

    2 Sec x + 1 = -1

    Sec x = -1

    Cos x = -1

    x =

    When you have a problem that requires you to square both sides, you must check your answer when you are done!


    Solving trig equations27
    Solving Trig Equations

    Sec x + 1 = Tan x

    x =


    Solving trig equations28
    Solving Trig Equations

    (Cos x + 1)² = Sin² x

    Cos x + 1 = Sin x

    Cos²x + 2Cos x + 1 = 1 – Cos² x

    2Cos² x + 2 Cos x = 0

    Cos x (2 Cos x + 2) = 0

    Cos x = 0

    Cos x = - 1

    x =

    x =


    Solving trig equations29
    Solving Trig Equations

    Cos x + 1 = Sin x

    x =



    Solving trig equations30
    Solving Trig Equations

    • Equations involving multiply angles

    • Solve the equation for the angle as your normally would

    • Then divide by the leading coefficient


    Solving trig equations31
    Solving Trig Equations

    • Solve the following trig equation for all values of x.

    2Sin 2x + 1 = 0

    2Sin 2x = -1

    Sin 2x = - ½

    2x =

    2x =

    x =

    x =


    Solving trig equations32
    Solving Trig Equations

    Redundant

    Answer


    Solving trig equations33
    Solving Trig Equations

    • Solve the following equations for all values of x.

    • 2Cos 3x – 1 = 0

    • Cot (x/2) + 1 = 0


    Solving trig equations34
    Solving Trig Equations

    2Cos 3x - 1 = 0

    2Cos 3x = 1

    Cos 3x = ½

    3x =

    3x =

    x =

    x =



    Solving trig equations36
    Solving Trig Equations

    • Topics covered in this section:

      • Solving basic trig equations

        • Finding solutions in [0, 2π)

        • Find all solutions

      • Solving quadratic equations

      • Squaring both sides and solving

      • Solving multiple angle equations

      • Using inverse functions to generate answers


    Solving trig equations37
    Solving Trig Equations

    Find all solutions to the following equation:

    Sec²x – 3Sec x – 10 = 0

    (Sec x + 2) (Sec x – 5) = 0

    Sec x + 2 = 0 Sec x – 5 = 0

    Sec x = 5

    Sec x = -2

    Cos x =

    Cos x = - ½

    x =

    x =


    Solving trig equations38
    Solving Trig Equations

    • One of the following equations has solutions and the other two do not. Which equations do not have solutions.

    • Sin²x – 5Sin x + 6 = 0

    • Sin²x – 4Sin x + 6 = 0

    • Sin²x – 5Sin x – 6 = 0

      Find conditions involving constants b and c that

      will guarantee the equation Sin²x + bSin x + c = 0

      has at least one solution.


    Solving trig functions
    Solving Trig Functions

    • Find all solutions of the following equation in the interval [0, 2π)

      Sec²x – 2 Tan x = 4

    1 + Tan²x – 2Tan x – 4 = 0

    Tan²x – 2Tan x – 3 = 0

    (Tan x + 1) (Tan x – 3) = 0

    Tan x = 3

    Tan x = -1


    Solving trig functions1
    Solving Trig Functions

    Tan x = -1

    Tan x = 3

    x = ArcTan 3

    ref. angle:

    71.6º

    I, III

    Quad:

    x =

    71.6º,

    251.6º


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