This presentation is the property of its rightful owner.
1 / 44

# 5.3 Solving Trig equations PowerPoint PPT Presentation

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

5.3 Solving Trig equations

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

## 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

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

### 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 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 Equations

Cos x = ½

2) Find the reference angle

x =

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

I =

IV =

### Solving Trig Equations

Cos x = ½

4) Create a formula if necessary

x =

x =

### 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

Sin x = - ½

Ref. Angle:

III:

Iv:

### Solving Trig Equations

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

Tan²x – 3 = 0

Tan²x = 3

Tan x =

Tan x =

Ref. Angle:

I:

III:

IV:

II:

x =

### 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 Equations

• Find all solutions to the following equation:

Sin x + = - Sin x

2 Sin x = -

x =

Sin x = -

x =

3Tan² x – 1 = 0

x =

Tan² x =

x =

Tan x =

x =

x =

### 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 =

## 5.3 Solving Trig equations

### 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 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 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 Equations

2Cos x – 1 = 0 Cos x + 1 = 0

Cos x = ½

Cos x = -1

Ref. Angle:

x =

I, IV

x =

### 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 Equations

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

Sin x = - ½

Sin x = 1

Ref. Angle:

x =

III, IV

x =

### 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 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 =

I, II

x =

### 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 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 =

I, IV

x =

### 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 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 Equations

Sec x + 1 = Tan x

x =

### 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 Equations

Cos x + 1 = Sin x

x =

## 5.3 Solving Trig equations

### 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 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 =

Redundant

### Solving Trig Equations

• Solve the following equations for all values of x.

• 2Cos 3x – 1 = 0

• Cot (x/2) + 1 = 0

### Solving Trig Equations

2Cos 3x - 1 = 0

2Cos 3x = 1

Cos 3x = ½

3x =

3x =

x =

x =

### Solving Trig Equations

• Topics covered in this section:

• Solving basic trig equations

• Finding solutions in [0, 2π)

• Find all solutions

• Squaring both sides and solving

• Solving multiple angle equations

• Using inverse functions to generate answers

### 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 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

• 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

Tan x = -1

Tan x = 3

x = ArcTan 3

ref. angle:

71.6º

I, III