Trigonometry Part III

8/6/2019 Trigonometry Part III

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Precalculus

Trigonometry III

By Dr. Dalia M. Gil, Ph.D.


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Topics

Periodic Functions.

Basic Identities.

Pythagorean Identities.

Sum and Difference Identities.

Confunction Identities.

Double Identities.


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Periodic Function

A function fis said to be periodic if thereexists a positive constant psuch that

f(s + p)= f(s)

for all sin the domain of f. The smallest

such positive number pis called theperiod of a function.


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Periodic Function f(s + p)= f(s)

sin (s + 2)= sin s and cos (s + 2)= cos s


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Basic Identities


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The equation of a unit circle in the xy-plane is x2 + y2 = 1

x = cos s and y = sin s

Pythagorean Identities

x2 + y2 = 1

(cos s)2 + (sin s)2 = 1

cos2 s + sin2 s = 1


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From sin2 s + cos2 s = 1, we can find otheridentities

Dividing both sidesby sin2 s


sin2 s + cos2 s = 1

sin2 s / sin2 s + cos2 s / sin2 s = 1 / sin2 s

1 + cot2 s = csc2 s Simplifying


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From sin2 s + cos2 s = 1, we can find otheridentities

Dividing both sidesby cos2 s


sin2 s + cos2 s = 1

sin2 s / cos2 s + cos2 s / cos2 s = 1 / cos2 s

tan2 s + 1 = sec2 s Simplifying


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sin2 s + cos2 s = 1

tan2 s + 1 = sec2 s

1 + cot2 s = csc2 s


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cos x (tan x sec x)

Example: multiply and simplify:

Simplifying Trigonometric Expressions

= cos x tan x cos x sec x

= cos x (sin x / cos x) cos x (1 / cos x)

= sin x 1


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sin2 x cos2 x + cos4 x



= cos2 x (sin2 x + cos2 x)

= cos2 x sin2 x + cos2 x = 1

Common factor


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cot (- ) / csc (- )



= [cos(- ) / (sin(- ) ] / [1 / sin(- )]

= cos (- ) = cos

= [cos(- ) / (sin(- ) ] . [sin(- )]


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[2 sin2 t + sin t 3] / [1 cos2 t sin t]



sin2 x + cos2 x = 1,or sin2 x = 1 - cos2 x

= [2 sin2

t + sin t 3] / [sin2

t sin

t ]

= (2 sin t + 3)(sin t 1) / sin t (sin t 1) ]

= (2 sin t + 3) / sin t


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Example: Multiply and simplify:



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Example: Rationalize the denominator:



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Example: Express (9 + x2) as atrigonometric function of without usingradicals by letting x = 3 tan .

Assume 0 < < /2.

Then find sin and cos



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Some important identities involving sumsor difference of two numbers or angles.

These numbers are identifies with theletters u and v.

Now we start for the identity cosine.

Sum and Difference Identities


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Lets consider a realnumber u in theinterval [/2, ] and areal number v in theinterval [0, /2].

These determine the

points A and B on theunit circle.



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The arc length sis

u v, and 0 s.

The coordinates of Aare (cos u, sin u),and the coordinate of

Bare (cos v, sin v).



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Using the formula ofdistance AB:



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Rotate the circle sothat point B is at (1,0).

Although thecoordinates of point Aare now (cos s, sin s),the distance AB has

not changed:



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Simplifying:



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Now we have two expressions for AB


Equating the both expressions for AB

Solving this expressions we obtain sum anddifference identities


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sin (u + v) = sin u cos v + cos u sin v. sin (u - v) = sin u cos v - cos u sin v.

cos (u + v) = cos u cos v - sin u sin v. cos (u - v) = cos u cos v + sin u sin v.

tan (u + v) = (tan u + tan v) / (1 - tan u tan v)

tan (u - v) = (tan u - tan v) / (1 + tan u tan v)



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Express as a difference of two numberswhose sine and cosine values are known:

Example: Find cos(5/12) exactly


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Find cos(5/12) exactly: Using cos (u - v) = cos u cos v + sin u sin v

Example: Sum and Difference Identities


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Express 105 as the sum of two measures:

105 = 45 + 60

sin 105 = sin (45 + 60)= sin 45 . sin 60 + cos 45 . cos 60

= 2/2 . + 2/2 . 3/2

= (2 + 6) / 4

sin 105 0.9659 and (2 + 6) / 4 0.9659

Example: Find sin 105 exactly


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Rewrite 15 as 45 - 30 and use the identityfor the tangent of a difference:

tan 15 = tan(45 - 30)= (tan 45 - tan 30) / (1 + tan 45 tan 30)

= ( 1 - 3/2) / (1 + 1 . 3/3)

= (3 - 3) / (3 + 3)

Example: Find tan 15 exactly


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Confunction Identities


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Prove the identity sin( + /2) = cos

sin(

+

/2) = sin

cos

/2 + cos

sin

/2

= sin . 0 + cos . 1

= cos

Example: Confunction Identities


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Other Confunction Identities


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Prove the identity sin( + /2) = cos

sin(

+

/2) = sin

cos

/2 + cos

sin

/2

= sin . 0 + cos . 1

= cos


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Find an identity for:

tan (x + /2)

= sin (x + /2) / cos (x + /2)

= cos x / - sin x

= - cot x



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Find an identity for:

sec (x 90)

= 1 / (cos (x 90)

= 1 / sin x

= csc x



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Double Identities


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To find these identities we will use the sumformulas to develop sin 2x

sin (u + v) = sin u cos v + cos u sin v.

Substitute x for both u and v

sin (x + x) = sin 2x

= sin x cos x + cos x sin x= 2 sin x cos x

Double-Angle Identities


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To find these identities we will use the sumformulas to develop cos 2x

cos (u + v) = cos u cos v - sin u sin v.


cos (x + x) = cos 2x

= cos x cos x - sin x sin x= cos2 x sin2 x



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To find these identities we will use the sumformulas to develop tan 2x

tan (u + v) = (tan u + tan v) / (1 - tan u tan v).


tan (x + x) = tan 2x

= (tan x + tan x) / (1 - tan x tan x).

= 2tan x / (1 tan2 x).



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Given tan = -3/4 and is in the quadrant II.Find each of the following

a) sin 2

b) cos 2c) tan 2d) The quadrant in which 2 lies.

sin = 3/5 and cos = -4/5


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a) sin 2= 2 sin cos = 2 . 3 / 5 . (-4 / 5) = -24 / 25

b) cos 2= cos2 - sin2= (-4 / 5)

2

(3/5)2

= 16 / 25 9 / 25

= 7 / 25


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c) tan 2= 2 tan / (1 tan2)= 2 (- 3 / 4 ) / ( 1 (- 3 / 4 )2

= (- 3 / 2) / (1 9/16)= - 24 / 7

d) sin 2

is negative and cos 2

is positive, so2 is in quadrant IV.


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Other Identities


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Half-Angle Identities

These identities are found taking squareroots and replacing x by x/2 .


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Example: Half-Angle Identities

Find tan(/8) exactly.


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References

Precalculus Graphs & Models. by M. L.

Bittinger et al. Addison Wesley Prentice1996 (pp 321 458 ).

Trigonometry Part III

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