Chapter 5 Trigonometric Equations€¦ · 11/5/2013 · 5/10/13 Audacity Obj: SWBAT simplify trig...
Transcript of Chapter 5 Trigonometric Equations€¦ · 11/5/2013 · 5/10/13 Audacity Obj: SWBAT simplify trig...
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MATHPOWERTM 12, WESTERN EDITION
5.4
5.4.1
Chapter 5 Trigonometric Equations
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5/10/13 Audacity
Obj: SWBAT simplify trig identities Trigonometric identities help you solve equations that arise when studying geometry, and in all kinds of higher
mathematics. I don't know what kind of level your mathematics is at, but rest assured that they will come in handy if
you do any senior or university level mathematics.
Trigonometry comes up in many places because of its links with geometry, examples are architecture, engineering,
even medicine. Trig identities also come up alot in physics and pure mathematics because Trig functions are the
simplest and most natural kinds of waves, and sometimes to analyse these ways we need to transform them into a
different form. Here trig identities come into play. Waves come up absolutely everywhere including physics, pure
mathematics, chemistry, sound engineering, civil engineering, aeronatics, they're just everywhere!
Bell Ringer: Simplify sinx cosx tanx cscx
HW Requests:
Pg 395 #72-75, 79, 80
Homework:
WS 14.3 Simplify Trig Identities
Education is Power!
Dignity without compromise!
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A trigonometric equation is an equation that involves
at least one trigonometric function of a variable. The
equation is a trigonometric identity if it is true for all
values of the variable for which both sides of the
equation are defined.
Trigonometric Identities
Prove that tan sin
cos.
y
x
y
r
x
r
y
r
r
x
y
x
L.S. = R.S. 5.4.2
Recall the basic
trig identities:
sin y
r
cos x
r
tan y
x
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5.4.3
Basic Trigonometric Identities
Quotient Identities
tan sin
coscot
cos
sin
Reciprocal Identities
sin 1
csccos
1
sectan
1
cot
Pythagorean Identities
sin2 + cos2 = 1 tan2 + 1 = sec2 cot2 + 1 = csc2
sin2 = 1 - cos2
cos2 = 1 - sin2
tan2 = sec2 - 1 cot2 = csc2 - 1
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5
Trigonometric identities
• sin2A + cos2A = 1
• 1 + tan2A = sec2A
• 1 + cot2A = cosec2A
• sin(A+B) = sinAcosB + cosAsin B
• cos(A+B) = cosAcosB – sinAsinB
• tan(A+B) = (tanA+tanB)/(1 – tanAtan B)
• sin(A-B) = sinAcosB – cosAsinB
• cos(A-B)=cosAcosB+sinAsinB
• tan(A-B)=(tanA-tanB)(1+tanAtanB)
• sin2A =2sinAcosA
• cos2A=cos2A - sin2A
• tan2A=2tanA/(1-tan2A)
• sin(A/2) = ±{(1-cosA)/2}
• Cos(A/2)= ±{(1+cosA)/2}
• Tan(A/2)= ±{(1-cosA)/(1+cosA)}
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Identities can be used to simplify trigonometric expressions.
Simplifying Trigonometric Expressions
cos sin tan
cos sin
sin
cos
cos
sin2
cos
cos 2 sin2
cos
1
cos
sec
a)
Simplify.
b) cot2
1 sin2
cos 2
sin2 cos
2
1
1
sin2
csc2
5.4.5
cos 2
sin2
1
cos2
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5.4.6
Simplifing Trigonometric Expressions
c) (1 + tan x)2 - 2 sin x sec x
1 2 tanx tan2x 2
sinx
cosx
1 tan2x 2tanx 2 tanx
sec2x
d) cscx
tan x cot x
1
sinx
sinx
cos x
cosx
sinx
1
sinx
sin2x cos
2x
sinxcos x
1
sinx
sinx cos x
1
cos x
1
sinx
1
sinx cos x
(1 tanx)2
2 sinx1
cosx
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End here 5/10
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sinA cos A 2
sin2A 2sinAcos A cos
2A
1 2sinAcos A
cos A
sinA1
sinA
cos A
sinA
sinA
1
= cosA
Trigonometric Identities [cont’d] Start with pg 446 Ex 1
5.4.4
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Trigonometric Identities
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Trigonometric Identities
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Trigonometric Identities
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Trigonometric Identities
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Trigonometric Identities
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Trigonometric Identities
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Trigonometric Identities
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Trigonometric Identities
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Trigonometric Identities
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Trigonometric Identities
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Trigonometric Identities
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Trigonometric Identities
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Trigonometric Identities
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Trigonometric Identities
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Trigonometric Identities
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Trigonometric Identities
** just combined fractions
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Trigonometric Identities
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Trigonometric Identities
Squared both sides…
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Trigonometric Identities
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Trigonometric Identities
Square root of both sides…
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5.4.7
Proving an Identity
Steps in Proving Identities
1. Start with the more complex side of the identity and work
with it exclusively to transform the expression into the
simpler side of the identity.
2. Look for algebraic simplifications:
• Do any multiplying , factoring, or squaring which is
obvious in the expression.
• Reduce two terms to one, either add two terms or
factor so that you may reduce. 3. Look for trigonometric simplifications:
• Look for familiar trig relationships.
• If the expression contains squared terms, think
of the Pythagorean Identities. • Transform each term to sine or cosine, if the
expression cannot be simplified easily using other ratios.
4. Keep the simpler side of the identity in mind.
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22 sincoscosecsin Establish the following identity:
In establishing an identity you should NOT move things
from one side of the equal sign to the other. Instead
substitute using identities you know and simplifying on
one side or the other side or both until both sides match.
22 sincoscosecsin
Let's sub in here using reciprocal identity
22 sincossin
1sin
22 sincos1
We often use the Pythagorean Identities solved for either sin2 or cos2.
sin2 + cos2 = 1 solved for sin2 is sin2 = 1 - cos2 which is our
left-hand side so we can substitute.
22 sinsin
We are done!
We've shown the
LHS equals the
RHS
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cos1
sincotcosec
Establish the following identity:
Let's sub in here using reciprocal identity and quotient
identity
Another trick if the
denominator is two terms
with one term a 1 and the
other a sine or cosine,
multiply top and bottom of
the fraction by the
conjugate and then you'll
be able to use the
Pythagorean Identity on
the bottom
We worked on
LHS and then
RHS but never
moved things
across the = sign
cos1
sincotcosec
cos1
sin
sin
cos
sin
1
cos1
sin
sin
cos1
combine fractions
cos1
cos1
cos1
sin
sin
cos1
2cos1
cos1sin
sin
cos1
FOIL
denominator
2sin
cos1sin
sin
cos1
sin
cos1
sin
cos1
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5.4.8
Proving an Identity
Prove the following:
a) sec x(1 + cos x) = 1 + sec x
= sec x + sec x cos x
= sec x + 1
1 + sec x
L.S. = R.S.
b) sec x = tan x csc x
sinx
cos x
1
sinx
1
cos x
secx
secx
L.S. = R.S.
c) tan x sin x + cos x = sec x
sinx
cos x
sinx
1 cosx
sin2 x cos 2 x
cos x
1
cos x
secx
secx
L.S. = R.S.
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d) sin4x - cos4x = 1 - 2cos2 x
= (sin2x - cos2x)(sin2x + cos2x)
= (1 - cos2x - cos2x)
= 1 - 2cos2x
L.S. = R.S.
1 - 2cos2x
e)
1
1 cos x
1
1 cosx 2 csc
2x
(1 cos x) (1 cosx)
(1 cosx)(1 cos x)
2
(1 cos2
x)
2
sin2x
2csc2x
2csc2x
L.S. = R.S.
Proving an Identity
5.4.9
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Proving an Identity
5.4.10
f)
cos A
1 sinA
1 sinA
cos A 2 secA
cos 2 A (1 sinA)(1 sinA)
(1 sinA)(cos A)
cos 2 A (1 2sinA sin2 A)
(1 sinA)(cos A)
cos 2 A sin2 A 1 2sinA
(1 sinA)(cos A)
2 2sinA
(1 sinA)(cos A)
2(1 sinA)
(1 sinA)(cos A)
2
(cos A)
2secA
2secA
L.S. = R.S.
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Using Exact Values to Prove an Identity
5.4.11
Consider sinx
1 cos x
1 cosx
sinx.
b) Verify that this statement is true for x =
6.
a) Use a graph to verify that the equation is an identity.
c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions.
y 1 cos x
sinxy
sinx
1 cos xa)
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sinx
1 cos x
1 cosx
sinx
1
2
1 3
2
b) Verify that this statement is true for x =
6.
sin
6
1 cos
6
1
2
2
2 3
1
2 3
1 cos
6
sin
6
1 3
2
1
2
2 3
2
2
1
2 3
2 3
1
2 3
2 3
2 3
2 3
4 3
2 3
Rationalize the
denominator:
1
2 3
L.S. = R.S.
Using Exact Values to Prove an Identity [cont’d]
5.4.12
Therefore, the identity is
true for the particular
case of x
6.
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c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions.
Using Exact Values to Prove an Identity [cont’d]
5.4.13
sinx
1 cos x
1 cosx
sinx
sinx
1 cos x
1 cos x
1 cos x
sinx(1 cosx)
1 cos2
x
sinx(1 cosx)
sin2x
1 cos x
sinx
1 cos x
sinx
L.S. = R.S.
Note the left side of the
equation has the restriction
1 - cos x ≠ 0 or cos x ≠ 1.
Therefore, x ≠ 0 + 2 n, where n is any integer.
The right side of the
equation has the restriction
sin x ≠ 0. x = 0 and Therefore, x ≠ 0 + 2n
and x ≠ + 2n, where n is any integer.
Restrictions:
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Proving an Equation is an Identity
Consider the equation sin2 A 1
sin2
A sinA 1
1
sinA.
b) Verify that this statement is true for x = 2.4 rad.
a) Use a graph to verify that the equation is an identity.
c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions.
y sin2 A 1
sin2
A sinAy 1
1
sinA
a)
5.4.14
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b) Verify that this statement is true for x = 2.4 rad.
Proving an Equation is an Identity [cont’d]
sin2 A 1
sin2
A sinA 1
1
sinA
(sin 2.4)2 1
(sin 2.4)2
sin2.4
= 2.480 466
1
1
sin 2.4
= 2.480 466
Therefore, the equation is true for x = 2.4 rad.
L.S. = R.S.
5.4.15
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5.4.16
Proving an Equation is an Identity [cont’d]
sin2 A 1
sin2
A sinA 1
1
sinA
c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions.
(sinA 1)(sinA 1)
sinA(sinA 1)
(sinA 1)
sinA
sinA
sinA
1
sinA
1 1
sinA
1 1
sinA
L.S. = R.S.
Note the left side of the
equation has the restriction:
sin2A - sin A ≠ 0
A 0, or A
2
Therefore, A 0 2 n or
A + 2n, or
A
2 2 n, where n is
any integer.
The right side of the
equation has the restriction
sin A ≠ 0, or A ≠ 0.
Therefore, A ≠ 0, + 2 n, where n is any integer.
sin A(sin A - 1) ≠ 0
sin A ≠ 0 or sin A ≠ 1