5.3 solving trigonometric equations
DESCRIPTION
5.3 Solving Trigonometric Equations. JMerrill , 2010. Recall (or Relearn ). It will be imperative that you know the identities from Section 5.1. Concentrate on the reciprocal, quotient, and Pythagorean identities. The Pythagorean identities are crucial!. Solve Using the Unit Circle. - PowerPoint PPT PresentationTRANSCRIPT
5.3Solving Trigonometric
Equations
JMerrill, 2010
It will be imperative that you know the identities from Section 5.1. Concentrate on the reciprocal, quotient, and Pythagorean identities.
The Pythagorean identities are crucial!
Recall (or Relearn )
Solve Using the Unit Circle Solve sin x = ½ Where on the circle does the sin x = ½ ?
5,6 6x
52 , 26 6x n n
Particular Solutions
General solutions
Solve for [0,2π]
Find all solutions
Solving a Trigonometric Equation Using Algebra
22sin 1 0 [0 ,360 )o oSolve for 22sin 1 0
22sin 1 2 1sin
2
2sin2
4 sin
2 2 .
There are solutionsbecause is positivein quadrants andnegative in quadrants
45 ,135 ,225 ,315o o o o
Find all solutions to: sin x + = -sin x
Using Algebra Again2
sinx sinx 2 0
2sinx 2
2sinx 2
5 7x 2n and x 2n4 4
Solve
You Try23tan x 1 0 f or [0,2 ]
3tanx 3
5 7 11x , , ,6 6 6 6
Solve by Factoring
sin tan 3 [0, )n 2six x xSolve for
sin tan 3sin 0x x x sin (tan 3) 0x x sin 0 tan 3x or x
Round to nearest hundredth
0,3.14 1.25,4.39x x
Solve
You Try2cot xcos x 2cot x in [0,2 )
2cot xcos x 2cot x 0 2cot x(cos x 2) 0
cot x 03x ,2 2
2
2
cos x 2 0cos x 2cosx 2DNE (Does Not Exist)No solution
Verify graphically
These 2 solutions are true because of the interval specified. If we did not specify and interval, you answer would be based on the period of tan x which is π and your only answer would be the first answer.
Quick review of Identities
Day 2 on 5.3
Fundamental Trigonometric Identities
Reciprocal Identities
1cscsin
1seccos
1cottan
Also true:
1sincsc
1cossec
1tancot
Fundamental Trigonometric Identities
Quotient Identities
sintancos
coscotsin
Fundamental Trigonometric Identities
Pythagorean Identities2 2sin cos 1
2 2tan 1 sec
These are crucial!You MUST know
them.2 21 cot csc
Pythagorean Memory Trick
sin2 cos2
tan2 cot2
sec2 csc2
(Add the top of the triangle to = the bottom)
1
Sometimes You Must Simplify Before you Can Solve
Strategies Change all functions to sine and cosine (or at
least into the same function) Substitute using Pythagorean Identities Combine terms into a single fraction with a
common denominator Split up one term into 2 fractions Multiply by a trig expression equal to 1 Factor out a common factor
Recall:Solving an algebraic equation
2 3 4 0( 1)( 4) 0( 1) 0 ( 4) 0 1 4
x xx xx or xx x
Solve
2 2sin sin cosx x x Hint: Make the words match so use a Pythagorean identity2 2sin sin 1 sinx x x
Quadratic: Set = 02 2sin sin 1 sin 0x x x Combine like
terms22sin sin 1 0x x Factor—(same as 2x2-x-
1)(2sin 1)(sin 1) 0x x
1sin sin 12
x or x 7 11, ,
2 6 6x
Solve2sin cos [0 ,360 )o oSolve for
2sin cos cos2sin
2 cot1 tan2
26.6 ,206.6o o
You cannot divide both sides by a common factor, if the factor cancels out. You will lose a root…
What You CANNOT Do
Example
2sin cos cos2sin
2 cot1 tan2
sin tan 3sinsin tan 3sin
sin sin tan 3
x x xx x xx xx
Common factor—lost a root
No common factor = OK
Sometimes, you must square both sides of an equation to obtain a quadratic. However, you must check your solutions. This method will sometimes result in extraneous solutions.
Squaring and Converting to a Quadratic
Solve cos x + 1 = sin x in [0, 2π) There is nothing you can do. So, square
both sides (cos x + 1)2 = sin2x cos2x + 2cosx + 1 = 1 – cos2x 2cos2x + 2cosx = 0 Now what?
Squaring and Converting to a Quadratic
Remember—you want the words to match so use a Pythagorean substitution!
2cos2x + 2cosx = 0 2cosx(cosx + 1) = 0 2cosx = 0 cosx + 1 = 0 cosx = 0 cosx = -1
Squaring and Converting to a Quadratic
3, 2 2
x x
3, 2 2
x x
Check Solutions
cos 1 sin2 2
0 1 1
3 3cos 1 sin2 2
0 1 1
cos 1 sin1 1 0
Solve 2cos3x – 1 = 0 for [0,2π) 2cos3x = 1 cos3x = ½ Hint: pretend the 3 is not there and solve
cosx = ½ . Answer:
But….
Functions With Multiple Angles
1 1cos2
5,3 3
x
x
Functions With Multiple Angles In our problem 2cos3x – 1 = 0 What is the 2? What is the 3? This graph is happening 3 times as often as
the original graph. Therefore, how many answers should you have?
amplitudefrequency
6
Functions With Multiple Angles
1 1cos2
5,3 3
x
x
Add a whole circle to each of these 7 11,3 3
And add the circle once again.
13 17,3 3
Functions With Multiple Angles
5 7 11 13 173 , , , , ,3 3 3 3 3
5 7 11 13 17, , , , ,9 9 9
,9 9
3
9
x
So x
Final step: Remember we pretended the 3 wasn’t there, but since it is there, x is really 3x:
Work the problems by yourself. Then compare answers with someone sitting next to you.
Round answers: 1. csc x = -5 (degrees)
2. 2 tanx + 3 = 0 (radians)
3. 2sec2x + tanx = 5 (radians)
Practice Problems
o o191.5 ,348.5
2.16, 5.30
2.16, 5.30, .79, 3.93
4. 3sinx – 2 = 5sinx – 1
5. cos x tan x = cos x
6. cos2 - 3 sin = 3
Practice – Exact Answers Only (Radians) Compare Answers
7 11,6 6
3 5, , ,2 2 4 4
32