Right Triangles

Right Right TrianglesTriangles

The Trig RatiosThe Trig RatiosBrought to you by Moody Mathematics

Let’s Let’s review review some some vocabularvocabulary.y.

Moody Moody MathematicsMathematics

AA

BBCC

Hypotenuse

Hypotenuse


AA

BBCC Opposite Opposite Leg to ALeg to A

Opposite Opposite LegLeg


BBOpp

osite

O

ppos

ite

Leg

to B

Leg

to BOpposite Opposite LegLeg


BBAdja

cent

Leg

Ad

jace

nt L

eg

to A

to A

Adjacent Adjacent LegLeg

AA


BBAdjacent Leg Adjacent Leg to Bto B

Adjacent Adjacent LegLeg


Consider the Consider the right right triangles in triangles in this next this next slide:slide:


What can you What can you say about say about them?them?


They They are are similarsimilar

By By AAAA


They They have the have the same same acute acute angleangle

They They have the have the same same right right angleangle


All All right right triangles triangles having one having one acute angle acute angle the same are the same are similar.similar. Moody Moody

MathematicsMathematics

For example, all 45-For example, all 45-45-90 triangles are 45-90 triangles are similar.similar.45

45

45


The legs of a 45-45-The legs of a 45-45-90 triangle are in a 90 triangle are in a 1 to 1 ratio.1 to 1 ratio.

5

5 3

3

9

9


In a 45-45-90 In a 45-45-90 triangle the ratio: triangle the ratio: leg leg

hypotenuse hypotenuse 5 5 2

3 239 29

1 .70712

45

45 45


Also, all 30-60-90 Also, all 30-60-90 triangles are similar.triangles are similar.

6060


In a 30-60-90 triangle, In a 30-60-90 triangle, the ratio: leg the ratio: leg opposite the 30opposite the 30o o hypotenuse hypotenuse 1

2( .5)or

6

12

4

830

30


The ratio:The ratio:leg opposite the 60leg opposite the 6000 hypotenuse hypotenuse

60

60

5

3 3

5 3

6

10

3

3 .86602


We have names for We have names for the 3 most common the 3 most common ratios that we will ratios that we will form in right form in right triangles. triangles. Moody Moody


The names are:The names are: the Sine Ratio, the Sine Ratio,

the Cosine Ratio, the Cosine Ratio, the Tangent the Tangent

Ratio. Ratio. Moody Moody MathematicsMathematics

leg opposite AHypotenuse

leg adjacent to AHypotenuse

leg opposite Aleg adjacent to A

Sin Sin A=A=Cos Cos A=A=

Tan Tan A=A= Moody Moody


S O H – C A H – S O H – C A H – T O AT O A““Some Old Some Old Hippy Hippy Caught Caught Another Another Hippy Hippy Tripping Tripping On On Antacid”Antacid”


SSOOHH

ininpposite pposite ypotenuypotenusese


CCAAHH

ososdjacent djacent ypotenuypotenusese


TTOOAA

ananpposite pposite djacentdjacent


AA

Opposite Opposite Leg to ALeg to A

Sin A = Sin A = Hypotenuse

Hypotenuse

CC BB

CBAB


BBAdja

cent

Leg

Ad

jace

nt L

eg

to A

to A

Cos A =Cos A =AAHypotenuse

Hypotenuse

ACAB

CCMoody Moody MathematicsMathematics

AA

BBCCOpposite Opposite Leg to ALeg to A

Tan A=Tan A=Ad

jace

nt L

eg

Adja

cent

Leg

to

Ato

A

CBAC


Now let’s set up the Now let’s set up the three ratios for three ratios for angle B. angle B.


BBOpp

osite

O

ppos

ite

Leg

to B

Leg

to BSin B= Sin B=

Hypotenuse

Hypotenuse

CAAB

AA

CCMoody Moody MathematicsMathematics

BBAdjacent Leg Adjacent Leg to Bto B

Cos B =Cos B =Hypotenuse

Hypotenuse

CBAB

CC

AA


BBOpp

osite

O

ppos

ite

Leg

to B

Leg

to BTan B =Tan B =

CC

AA

Adjacent Leg Adjacent Leg to Bto B

ACCB


Now let’s use a ratio Now let’s use a ratio to solve for a to solve for a missing side of a missing side of a right triangle:right triangle:


AA

BBCC

12

27

x

Let’s estimate the Let’s estimate the value of x before we value of x before we start: a. X>12start: a. X>12

b. b. 6<x<126<x<12

c. c. X<6 X<6


AA

BBCC

12

27

x

a. a. X>12 X>12

b. b. 6<x<16<x<122

c. X<6 c. X<6

It’s not (a) because It’s not (a) because A leg can’t be A leg can’t be longer than the longer than the hypotenuse.hypotenuse.


AA

BBCC

12

27

x

b. b. 6<x<16<x<122

c. X<6 c. X<6

If B were 30If B were 30oo then x would then x would be 6 exactly. Since B is be 6 exactly. Since B is

smaller than smaller than 3030oo x<6. x<6.


AA

BBCC

12

27

x

Look at the parts Look at the parts involved and involved and decide which decide which ratio “fits” best. ratio “fits” best.


AA

BBCC

12

27

x

Where are the Where are the given and given and missing sides in missing sides in relation to the relation to the known angle?known angle?


AA

BBCC

12

27

x

X is the X is the “opposite leg” “opposite leg” to B and 12 is to B and 12 is the the “hypotenuse”.“hypotenuse”.


AA

BBCC

12

27

x

sin 2712x

12 sin 27 x

( )5.45calculator

x


Now let’s use Now let’s use another ratio to another ratio to solve for a missing solve for a missing side of a right side of a right triangle:triangle:


AA

BBCC

16

35

x

Let’s estimate the Let’s estimate the value of x before we value of x before we start: a. X>16start: a. X>16

b. b. 8<x<168<x<16

c. c. X<8 X<8


AA

BBCC

16

35

x

It’s not (a) because It’s not (a) because A leg can’t be A leg can’t be longer than the longer than the hypotenuse.hypotenuse.

a. a. X>16 X>16

b. b. 8<x<18<x<166

c. X<8 c. X<8 Moody Moody MathematicsMathematics

AA

BBCC

16

35

x

b. b. 8<x<18<x<166

c. X<8 c. X<8

If A were 30If A were 30oo then x would then x would be 8. Since <A =55be 8. Since <A =55oo is is

bigger thanbigger than3030o,o, x>8. x>8.


AA

BBCC

16

35

x



AA

BBCC

16

35

x

Where are the Where are the given and given and missing sides in missing sides in relation to the relation to the known angle?known angle?


AA

BBCC

16

35

x

X is the X is the “adjacent leg” “adjacent leg” to B and 16 is to B and 16 is the the “hypotenuse”.“hypotenuse”.


AA

BBCC

16

35

x

cos3516x

16 cos35 x

13.12( )calculator

x


Now let’s solve Now let’s solve another ratio to find another ratio to find a missing side of a a missing side of a right triangle, right triangle, but but this time x is on the this time x is on the bottom.bottom.


AA

BBCC 22

53

x



AA

BBCC 22

53

x

Notice that only Notice that only the legs are the legs are involved, not involved, not the hypotenuse. the hypotenuse.


AA

BBCC 22

53

x

22tan53x

1

tan53 22x

22tan53

x

16.58x


Now let’s solve a Now let’s solve a ratio to find a ratio to find a missing missing angleangle of a of a right triangle.right triangle.


AA

BBCC 9.1

x

12

Let’s estimate the Let’s estimate the value of x before we value of x before we start: a. X<45start: a. X<45oo

b. b. X> 45X> 45oo


AA

BBCC 9.1

x

12

a.a. X<45X<45oo

b. X> 45b. X> 45oo

If x were 45If x were 45oo then both then both legs would be (which legs would be (which is between 8 and 9). is between 8 and 9).

6 2


AA

BBCC 9.1

x

12



AA

BBCC 9.1

x

12

9.1 is the 9.1 is the “opposite leg” “opposite leg” to x and 12 is to x and 12 is the the “hypotenuse”.“hypotenuse”.


AA

BBCC 9.1

x

12

9.1sin12

x

1sin 9.1 12 x

49.3 x

Hit the [2Hit the [2ndnd] key ] key then [sin] keythen [sin] key


Now let’s solve Now let’s solve another ratio to find another ratio to find a missing a missing angleangle of of a right triangle.a right triangle.


AA

BBCC 15.8x

17



AA

BBCC 15.8x

17

15.8 is the 15.8 is the “adjacent leg” “adjacent leg” to x and 17 is to x and 17 is the the “hypotenuse”.“hypotenuse”.


AA

BBCC 15.8x

17

15.8cos17

x

1cos 15.8 17 x

21.7 x

Hit the [2Hit the [2ndnd] key ] key then [cos] keythen [cos] key


The EndThe End Now go practice!Now go practice!


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Right Triangles

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