analytic trigognometry
DESCRIPTION
Analytic TrigonometryTRANSCRIPT
TOPIC 1.2 – ANALYTIC TRIGONOMETRY
1.2.1: The Inverse Sine, Cosine, and Tangent Functions1.2.2: The Inverse Trigonometric Functions1.2.3: Trigonometric Identities1.2.4: Sum and Difference Formulas1.2.5: Double-Angle and Half-Angle Formulas1.2.6: Product-to-Sum and Sum-to-Product Formulas1.2.7: Trigonometric Equations (I)1.2.8: Trigonometric Equations (II)
1
2
Review of Properties of Functions and Their Inverses
1.2.1& 1.2.2: The Inverse Sine, Cosine, and Tangent Functions
3
1sin
xy sin22
x
y x x y sin sin1
The inverse sine function:The inverse sine function denoted by
sine function from . Thus,
is the inverse of the restricted
4
x1sin
x1sin
Finding exact values of
1. Let = xsin2. Rewrite = as x1sin
xsin3. Use the exact values to find the value of that satisfies
2
3sin 1
2
2sin 1
Example: Find the exact value of;1- 2-
5
6
1cos
xy cos x0
The inverse cosine function:The inverse cosine function denoted by restricted cosine function from . Thus,
is the inverse of the
y x x y cos cos1
0 y 1 1xwhere and
Example: Find the exact value of;1)
2
1cos 1
7
8
1tan
xy tan 2 2
y
The inverse tangent function:The inverse tangent function denoted by
restricted tangent function from .Thus,
is the inverse of the
y x x y tan tan1
2 2
y xwhere and
Example: Find the exact value of;1) 1tan 1
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Composition of functions involving inverse trigonometric functions
xx 1sinsin
xx sinsin 1
2,2
Inverse properties1.Sine function:
for every x in the interval [-1,1]
for every x in the interval
xx 1coscos
xx coscos 1 ,0
2. Cosine function:
for every x in the interval [-1,1]
for every x in the interval
xx 1tantan
xx tantan 1
2,2
3. Tangent function:for every real number x
for every x in the interval
10
7.0coscos 1 sinsin 1 2coscos 1
Example:1-Find the exact value if possible;a) b) c)
4
3tansin 1
2
1sincos 1d) e)
2- If x > 0, write x1tansec as an algebraic expression in x
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1.2.3: Trigonometric Identities
Fundamental trigonometric identities
i ii iii.cscsin
.seccos
.cottan
1 1 1
Reciprocal Identities
i ii. tansin
cos.cot
cos
sin
Quotient Identities
i ii
iii
.sin cos . tan sec
. cot csc
2 2 2 2
2 2
1 1
1
Pythagorean Identities
i ii iii
iv v vi
.sin( ) sin .cos( ) cos . tan( ) tan
.csc( ) csc .sec( ) sec .cot( ) cot
Even - Odd Identities
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Example: Verify the identity:Changing to sine and cosine1) xxx sectancsc 2) xxxx cscsincotcos
3) xxxx
xxcossin
cscsec
)csc(sec
Using factoring1) xxxx 32 sincossinsin 2) x
x
x
x
xcsc2
sin
cos1
cos1
sin
Multiplying numerator and denominator by the same factor1)
x
x
x
x
cos
sin1
sin1
cos
Working with both sides separately1)
2tan22sin1
1
sin1
1
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1.2.4: Sum and Difference Formulas
Sum and difference formulas for cosines and sines
cos (A+ B) = cos A cos B - sin A sin Bcos (A - B) = cos A cos B + sin A sin Bsin (A + B) = sin A cos B + cos A sin Bsin (A - B) = sin A cos B - cos A sin B
Example:1.Using difference formula to find the exact value
a) Give the exact value of 6090cos30cos using the sum and difference formula
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5sin
4612
5 b) Find the exact value of using the fact that
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tantan1coscos
cos
5
4sin angle
2
1sin
angle
2. Verify the identity:
3. Suppose that for a quadrant II and
for a quadrant I . Find the exact value ofcos cos
cos sin
a) b)
c) d)
15
tan( )tan tan
tan tan
1
tan( )tan tan
tan tan
1
Sum and difference formulas for tangents1-
2-
Example:1.Verify the identity: xx tantan
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1.2.5: Double-Angle and Half-Angle Formulas
cossin22sin
22 sincos2cos
2tan1
tan22tan
Double angle formulas:1- 2-
3-
5
4sin
Example:
1- If and lies in quadrant II, find the exact value of;
2sin 2cos 2tana) b) c)
2. Find the exact value of 15sin15cos 22
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Using Pythagorean identity to write 2cos in terms of sine only:
22 sincos2cos
1cos22cos 2 2sin212cos
Three forms of the double angle formula for cos
1-
2-
3-
Example: Verify the identity: 3sin4sin33sin
2
2cos1sin2
2
2cos1cos2
2cos1
2cos1tan2
Power reducing formulas
x4sin
of trigonometric functions greater than 1
that does not contain powers Example: Write an equivalent expression for
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Half angle formulas
cos1
cos1
2tan;
2
cos1
2cos;
2
cos1
2sin
The + or – in each formula is determined by the quadrant in which 2
lies
Example:1- Use cos 120º to find the exact value of cos 105º2- Verify the identity:
2cos1
2sintan
2
sin
cos1
2tan
cos1
sin
2tan
Half angle formula for tan
Example: Verify the identity:
csccscsec
sec
2tan
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1.2.6: Product-to-Sum and Sum-to-Product Formulas
)]cos()[cos(2
1sinsin
)]cos()[cos(2
1coscos
)]sin()[sin(2
1cossin
)]sin()[sin(2
1sincos
1-
2-
3-
4-
Example: Express each of the following products as a sum or difference:a. sin 5x sin 2x b. cos 7x cos x
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2cos
2sin2sinsin
2cos
2sin2sinsin
2cos
2cos2coscos
2sin
2sin2coscos
Sum to Product Formulas:1-
2-
3-
4-
Example:1- Express each sum or difference as a producta. sin 7x + sin 3x b. cos 3x +cos 2x
2- Verify the identity: xxx
xxtan
sin3sin
cos3cos
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1.2.7& 1.2.8: Trigonometric Equations
• A trigonometric equation is an equation that contains a trigonometric expression. • To solve an equation containing a single trigonometric function:
Isolate the function on one side of the equation Solve for the variable
Finding all solutions of a trigonometric equationExample: Solve the equation: 3sin3sin5 xx
Solving an equation with a multiple angleExample: Solve the equation:
32tan x
20 x
2
1
3sin x
20 x
1-
2-
22
01sin3sin2 2 xx 20 x
xxx sintansin 20 x
Trigonometric equations quadratic in form
Example: Solve the equation:
Using factoring to separate 2 different trigonometric functions in an equation
Example: Solve the equation:
0sin2cos xx 20 x
2
1cossin xx 20 x
1sincos xx 20 x
Using an identity to solve a trigonometric equation
Example: Solve the equation:1-
2-
3-
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