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626 Chapter 5 Analytic Trigonometry
Section 5.5 Trigonometric Equations
Objectives
�Find all solutions of a xponential functions display the manic
trigonometric equation. Eenergies of uncontrolled growth. By
contrast, trigonometric functions repeat
�Solve equations with their behavior. Do they embody in their
multiple angles. regularity some basic rhythm of the
�Solve trigonometric equations universe? The cycles of periodic phenomena
quadratic in form. provide events that we can comfortably
�Use factoring to separate count on.When will the moon look just as
different functions in it does at this moment? When can I count
trigonometric equations. on 13.5 hours of daylight? When will my
�Use identities to solve breathing be exactly as it is right now?
trigonometric equations. Models with trigonometric functions
embrace the periodic rhythms of our
�Use a calculator to solve world.Equations containing trigonometric
trigonometric equations. functions are used to answer questions
about these models.
�Find all solutions of a Trigonometric Equations and Their Solutions
trigonometric equation. A trigonometric equation is an equation that contains a trigonometric expression
with a variable, such as sin x. We have seen that some trigonometric equations are
2 2
identities, such as sin x + cos x = 1. These equations are true for every value of
the variable for which the expressions are defined. In this section, we consider
trigonometric equations that are true for only some values of the variable. The
values that satisfy such an equation are its solutions. (There are trigonometric
equations that have no solution.)
An example of a trigonometric equation is
sin x = 1.
2
p p 1
A solution of this equation is 6 because sin 6 = 2. By contrast, p is not a solution
because sin p = 0 Z 1.
2
Is p the only solution of sin x = 1? The answer is no. Because of the periodic
6 2 1
nature of the sine function,there are infinitely many values of x for which sin x = 2.
p 3p 7p
Figure 5.7 shows five of the solutions, including 6, for - 2 … x … 2 . Notice that
the x-coordinates of the points where the graph of y = sin x intersects the line
y = 1 are the solutions of the equation sin x = 1.
2 2
y y = sin x
y = 1 1
2
Figure 5.7 The equation sin x = 1 x
2 −' k l m x
has five solutions when x is restricted to
the interval c- 3p, 7pd. −1
2 2 solution solution solution solution solution
How do we represent all solutions of 1 Because the period of the sine
sin x = 2?
function is 2p, first find all solutions in 30, 2p2. The solutions are
x=p and x=p-p=5p.
6 6 6
The sine is positive in quadrants I and II.
Section 5.5 Trigonometric Equations 627
Any multiple of 2p can be added to these values and the sine is still 1. Thus, all
solutions of sin x = 1 are given by 2
2
x = p + 2np or x = 5p + 2np,
6 6
where n is any integer. By choosing any two integers, such as n = 0 and n = 1, we
can find some solutions of sin x = 1. Thus, four of the solutions are determined as
2
follows:
Let n = 0. Let n = 1.
p 5p p 5p
x=6+2 � 0p x= 6+2 � 0p x=6+2 � 1p x=6+2 � 1p
p 5p p 5p
=6 =6 =6+2p x=6+2p
p 12p 13p 5p 12p 17p
=6+6=6 =6+6=6 .
These four solutions are shown among the five solutions in Figure 5.7.
Equations Involving a Single Trigonometric Function
To solve an equation containing a single trigonometric function:
Isolate the function on one side of the equation.
Solve for the variable.
EXAMPLE 1 Finding All Solutions of a Trigonometric Equation
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 terms with sin x on the left side and constant terms on the right side.
3 sin x - 2 = 5 sin x - 1 This is the given equation.
3 sin x - 5 sin x - 2 = 5 sin x - 5 sin x - 1 Subtract 5 sin x from both sides.
-2 sin x - 2 = -1 Simplify.
-2 sin x = 1 Add 2 to both sides.
1
sin x = - Divide both sides by -2 and solve
2
for sin x.
Step 2 Solve for the variable. We must solve for x in sin x = - 1. Because
p 1 1 2
sin 6 = 2, the solutions of sin x = - 2 in 30, 2p2 are
p 6p p 7p p 12p p 11p
x=p+6=6+6=6 x=2p-6= 6-6= 6 .
The sine is negative The sine is negative
in quadrant III. in quadrant IV.
Because the period of the sine function is 2p, the solutions of the equation are
given by
x = 7p + 2np and x = 11p + 2np,
6 6
where n is any integer.
922 Chapter 7 | Trigonometric Identities and Equation
advantage of using the identities we developed in the previous sections.
General Strategy for solving trig equations
+2kπ [0, 2π)
Unit Circle
Example 7.45
Solving a Linear Trigonometric Equation Involving the Cosine Function
Find all possible exact solutions for the equation cos θ = 1.
2
Solution
Fromtheunit circle, we know that cosine is positive in QI and QIV. Cosine is an x value on the unit circle, so we
want the angles on the unit circle where x is 1/2. Let's find the the angles on the unit circle.
This OpenStax book is available for free at https://legacy.cnx.org/content/col26003/1.12
Chapter 7 | Trigonometric Identities and Equation 923
cos θ = 1
2
θ = π, 5π
3 3
[ ]
These are the solutions in the interval 0, 2π . All possible solutions are given by
π ±2kπ and 5π ±2kπ
3 3
where k is an integer.
Example 7.46
Solving a Linear Equation Involving the Sine Function
Find all possible exact solutions for the equation sin t = 1.
2
Solution
First we want to solve in one full cycle. We know sine is positive in QI and QII. Since sine is a y value we want
the angles in QI and QII whose y values are 1 .
2
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