Chapter 6 6.5 Trigonometric equations

1
Chapter 66.5 Trigonometric equations

• Find all solutions of a trigonometric equation.
• Solve equations with multiple angles.
• Solve trigonometric equations quadratic in form.
• Use factoring to separate different functions in
  trigonometric equations.
• Use identities to solve trigonometric equations.
• Use a calculator to solve trigonometric equations

• Dr .Hayk Melikyan
• Departmen of Mathematics and CS
• melikyan_at_nccu.edu

2
Trigonometric Equations and Their Solutions

• A trigonometric equation is an equation that
  contains a trigonometric expression with a
  variable, such as sin x.
• The values that satisfy such an equation are its
  solutions. (There are trigonometric equations
  that have no solution.)
• When an equation includes multiple angles, the
  period of the function plays an important role in
  ensuring that we do not leave out any solutions.

3
Equations Involving a Single Trigonometric
Function

• To solve an equation containing a single
  trigonometric function
• Isolate the function on one side of the
  equation.
• sinx a (-1 a 1 )
• cosx a (-1 a 1 )
• tan x a ( for any real a )
• Solve for the variable.

4
Trigonometric Equations
y
y
cos
x
1
y
0.5
x
4
?
2
?
2
?
4
?
1
cos x 0.5 has infinitely many solutions for ?
lt x lt ?
y
y
cos
x
1
0.5
2
?
cos x 0.5 has two solutions for 0 lt x lt 2?
1
5
Text Example
Solve the equation 3 sin x - 2 5 sin x - 1.
Solution The equation contains a single
trigonometric function, sin x.
Step 1 Isolate the function on one side of the
equation. We can solve for sin x by collecting
all terms with sin x on the left side, and all
the constant terms on the right side.
6
Text Example
Solve the equation 2 cos2 x cos x - 1 0, 0
x lt 2p.
cos x 1/2
x p????? x 2p??p?????p????? x p???
The solutions in the interval 0, 2p) are p/3, p,
and 5p/3.
7
Example

• Solve the following equation

Solution
8
Example

• Solve the equation on the interval 0,2?)

Solution
9
Example

• Solve the equation on the interval 0,2?)

Solution
10
Example

• Solve the equation on the interval 0,2?)

Solution
11
Example Finding all Solutions of a
Trigonometric Equation

• Solve the equation
• Step 1 Isolate the function on one side of the
  equation.

12
Example Finding all Solutions of a
Trigonometric Equation (continued)

• Solve the equation
• Step 2 Solve for the variable.

Solutions for this equation in are
The solutions for this equation are
13
Solving an Equation with a Multiple Angle

• Solve the equation

Because the period is all solutions for
this equation are given by
14
Solving an Equation with a Multiple Angle

• Solve the equation

Because the period is all solutions for this
equation are given by
In the interval , the solutions are
15
Solving a Trigonometric Equation Quadratic in
Form

• Solve the equation

The solutions in the interval for this equation
are
16
Using Factoring to Separate Different Functions

• Solve the equation

The solutions for this equation in the interval
are
17
Using an Identity to Solve a Trigonometric
Equation

• Solve the equation

The solutions in the interval are
18
Solving Trigonometric Equations with a Calculator

• Solve the equation, correct to four decimal
  places, for

tanx is positive in quadrants I and III
In quadrant I
In quadrant III
The solutions for this equation are 1.2592 and
4.4008.
19
Using a Calculator to Solve Trigonometric
Equations

• Solve the equation, correct to four decimal
  places, for

Sin x is negative in quadrants III and IV
In quadrant III
In quadrant IV
The solutions for this equation are 3.3752 and
6.0496.