right angle trigonometry. 19 july 2011 alg2_13_01_rightangletrig.ppt copyrighted © by t. darrel...

Right Angle Trigonometry

19 July 2011 Alg2_13_01_RightAngleTrig.pptCopyrighted © by T. Darrel Westbrook

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– To find values of the six trigonometric functions for acute angles,

– To understand the two Special Trigonometric triangles, and

– To solve problems involving right triangles.

What You Will Learn:


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Definition: Trigonometry – is the study of the relationships among the angles and sides of a right triangle.


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Given : Angle AGiven : Angle B

Opposite Side

Adjacent Side

Labeling a Triangle

A

B

C

ac

b

Opposite Side

Adjacent Side

Hypotenuse


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What makes Trigonometry work?

Similar Right TrianglesWhat is required for two right triangles to be similar?


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ABAD

BCDE

CAEA

= =ABAD

BCDE

CAEA

= =

A

B

C

Given a right triangle

D

E

ABC ADE

ABAD

BCDE

=

DEAD

BCAB

=

Opposite SideDE

HypotenuseAD

Opposite SideBC HypotenuseAB

=

Opposite SideDE

HypotenuseAD=

Opposite SideBC

HypotenuseAB

Divide by AB and multiply by DE


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A

B

C

D

E

ABC ADE

ABAD

BCDE

CAEA

= =

ABAD

CAEA

=

EAAD

CAAB

=

Opposite SideDE

HypotenuseAD=

Opposite SideBC

HypotenuseAB

Adjacent SideEA

HypotenuseAD

Adjacent SideCA HypotenuseAB

=

Adjacent SideEA

HypotenuseAD=

Adjacent SideCA

HypotenuseAB

Divide by AB and multiply by EA


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A

B

C

D

E

ABC ADE

ABAD

BCDE

CAEA

= =

BCDE

CAEA

=

BCCA

DEEA

=

Opposite SideDE

HypotenuseAD=

Opposite SideBC

HypotenuseAB

Opposite SideBC

Adjacent SideCA

Opposite SideDE

Adjacent SideEA

=

Adjacent SideEA

HypotenuseAD=

Adjacent SideCA

HypotenuseAB

Opposite SideBC

Adjacent SideCA=

Opposite SideDE

Adjacent SideEADivide by BC and multiply by EA


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A

B

C

D

E

Opposite SideDE

HypotenuseAD=

Opposite SideBC

HypotenuseAB

Adjacent SideEA

HypotenuseAD=

Adjacent SideCA

HypotenuseAB

Opposite SideBC

Adjacent SideCA=

Opposite SideDE

Adjacent SideEA

No matter the length of the sides of the right triangle, these ratios remain equal for a given acute angle. So, what does this imply?


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So, for every right triangle with an acute angle A, the various ratios of

the opposite side, adjacent side, and the hypotenuse are the same, no matter the length of the sides of the triangle, as long as the angles are the same and

the triangles are similar.

Opposite SideDE

HypotenuseAD=

Opposite SideBC

HypotenuseAB

Adjacent SideEA

HypotenuseAD=

Adjacent SideCA

HypotenuseAB

Opposite SideBC

Adjacent SideCA=

Opposite SideDE

Adjacent SideEA

sin A =

cos A =

tan A =

A

B

C

D

E

Let’s call

These are the three basic trigonometric functions.


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Opposite SideBC

HypotenuseAB

Adjacent SideCA

HypotenuseAB

Opposite SideBC

Adjacent SideCA

sin A =

cos A =

tan A =

A

B

C

sin B =

cos B =

tan B =

Opposite SideCA

HypotenuseAB

Adjacent SideBC

HypotenuseAB

Opposite SideCA

Adjacent SideBC

= cos A

= sin A

1tan A

=

Each pair of equal trigonometric functions

are called co-functions of the acute angles of the

right triangle.

= cot A


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Definition of Six Basic Trig Functions

A

B

C

ac

b

Given : Angle A

sin A = Opposite SideHypotenuse

cos A = Adjacent SideHypotenuse

tan A = Opposite SideAdjacent Side

csc A = Hypotenuse

Opposite Side

sec A = Hypotenuse

Adjacent Side

cot A = Adjacent SideOpposite Side


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A mnemonic use to help remember the first three basic trigonometric functions is:

SOH-CAH-TOA

Sine Opp over Hyp

Cosine Adj over Hyp

Tangent Opp over Adj

The cosecant (csc) is the inverse of the sine.

The secant (sec) is the inverse of the cosine.

The cotangent (cot) is the inverse of the tangent.


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What do the graphs of these trigonometric functions look like?


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sin

The x-axis scale is –2 to 2. Note that it completes a cycle every 2

radians.

The y-axis scale is –1.5 to 1.5, but what is the maximum/minimum

value of the sine function?


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To find non-baseline periods divide the baseline by the coefficient. Example: sin 3. The non-baseline period is 2/3 or every 120. The baseline period for the cosecant

function is the same.

The number of radians a trig function requires to complete one cycle is called the function’s baseline period. The

baseline occurs when the coefficient for is 1. The sine’s baseline period is 2. Its domain is 0 to 2.

sin


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cos


radians.

The y-axis scale is –1.5 to 1.5, but what is the maximum/minimum value of the cosine

function?


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The cosine’s baseline period is 2.

Cosine domain is 0 to 2.

The baseline period for the secant function is the same.

cos


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tan

The x-axis scale is –2 to 2. Note that it completes a cycle every

radians.

The y-axis scale is –1.5 to 1.5, but what is the maximum/minimum value of the tangent

function?


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The tangent’s baseline period is .

Tangent domain is –/2 to /2.

The baseline period for the cotangent function is the same.

tan


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csc


radians.

The y-axis scale is –5 to 5, but what is the

maximum/minimum value of the cosecant

function?

Note the scale change.


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sec


radians.

The y-axis scale is –10 to 10, but what is the maximum/minimum value of the secant

function?


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cot

The x-axis scale is –2 to 2. Note that it completes a cycle every

radians.

The y-axis scale is –10 to 10, but what is the maximum/minimum

value of the cotangent function?


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sin and csc

The x-axis scale is –2 to 2. Note that they complete a cycle every 2 radians (right half of graph).


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cos and sec

The x-axis scale is –2 to 2. Note that they complete a cycle every 2 radians (right half of graph).


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tan and cot

The x-axis scale is –2 to 2. Note that they complete a cycle every 2 radians, and they are shifted /2 radians

from each other.


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A

B

DC

Given: Equilateral Triangle

Special Triangles

AC = AB/2

AB2 = AC2 + BC2

BC2 = AB2 – AC2

BC2 = AB2 – (AB/2)2

BC2 = AB2 – AB2/4

BC2 = (3/4) AB2

BC = (3/2) AB

Let x = AB

BC = (3/2) x

START


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A

B

DC

Given: Equilateral Triangle

Special Triangles

Let x = AB

BC = (3/2) xAC = x/2

x/2

x

(3/2 ) x

2w

w

w3

These relationships are true for any 30-

60o triangle

Which relationship you use depends on

the problem.


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For example: given a 30-60 triangle with the hypotenuse of length 10 units, what are the lengths of the other two

sides?

10

The largest side is across from which

angle?

60o

30o

10/2 = 5

53The 30-60 triangle relationship used

was:

x

x/2

x3/2


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Suppose we only knew the short side and its length is 9. What is the length of the other side and the hypotenuse?

9

2 9 = 1893

The 30-60 triangle relationship used

was:x

x32x

60o

30o


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Suppose we only knew the long side and its length is 7. What is the length of the other side and the hypotenuse?

7/3

2 7/3 = 14/3

7

x3 = 7

x = 7/3

60o

30o


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Given: Right Isosceles Triangle

Special Triangles

A

B

C

AC = BC

AB2 = AC2 + BC2

AB2 = AC2 + AC2

AB2 = 2AC2

AB = AC2

Let x = AC = BC

Then AB = x 2 x

x

x 2


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Given: Right Isosceles Triangle

Special Triangles

A

B

C

Another Form

Given AB = x

x

x 2

x 2


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B

3

7


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B

3

7


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The answer is D.

B

3

7


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What You Have Learned:

– To find values of trigonometric functions for acute angles, and

– To solve problems involving right triangles.


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END OF LINE


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right angle trigonometry. 19 july 2011 alg2_13_01_rightangletrig.ppt copyrighted © by t. darrel...

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