6.5. Trigonometric Equations http://www.ck12.org
Solve the following equation.
4 cos^2 x− 1 = 3 −4 sin^2 x
Solution:
4 cos^2 x− 1 = 3 −4 sin^2 x
4 cos^2 x+4 sin^2 x= 3 + 1
4 (cos^2 x+sin^2 x) = 4
4 = 4This equation is always true which means the right side is always equal to the left side. This is an identity.
Concept Problem Revisited
The equation cosx=1 has many solutions. When you type cos−^1 1 on your calculator, it will yield only one solution
which is 0. In order to describe all the solutions you must use logic and the graph to figure out that cosine also has a
height of 1 at− 2 π, 2 π,− 4 π, 4 π...Luckily all these values are sequences in a clear pattern so you can describe them
all in general with the following notation:
x= 0 ±n· 2 πwherenis an integer, orx=±n· 2 πwherenis an integer
Vocabulary
The terms“general solution,” “completely solve”, and“solve exactly”mean you must find solutions to an equation
without the use of a calculator. In addition, trigonometric equations may have an infinite number of solutions that
repeat in a certain pattern because they are periodic functions. When you see these directions remember to find all
the solutions by using notation like in Example B.
Guided Practice
- Solve the following equation on the interval( 2 π, 4 π).
2 sinx+ 1 = 0 - Solve the following equation exactly.
2 cos^2 x+3 cosx− 2 = 0 - Create an equation that has the solutions:
π 4 ±n· 2 πwherenis an integer
Answers:
- First, solve for the solutions within one period and then use logic to find the solutions in the correct interval.
2 sinx+ 1 = 0
sinx=−^12
x=^76 π,^116 πYou must add 2πto each of these solutions to get solutions that are in the interval.