2.3 The Sine and Cosine Ratios

MFM2P

Back in the day….

Yesterday, we created 3 similar triangles…

A

F

E

G

B C D

20 units 10 units 10 units

1015

20

E

B

20

10

A A C

30

15

F

A

G

D

20

40

Neat Things about Ratios!!! Looking at Ratio 1

G

E

B

20

10

A A C

30

15

F

A

G

D

20

40

BE

AE 10 .

22.3

CF

AF 15 .

33.5

DG

AF 20 .

44.7

= 0.45 = 0.45= 0.45

When I look at the triangle from A OPPOSITE .

HYPOTENUSE0.45

OPP

ADJ A

HYP

22.333.5

44.7

The Sine Ratio!

When I look at the triangle from A OPPOSITE .

HYPOTENUSE0.45

There is a special name for this ratio. It is called the SINE RATIO. We can use it to solve for A.

OPP

ADJ A

HYP

OPPOSITE . HYPOTENUSESIN A =

The Sine Ratio!

If we know A, and one side, we can calculate the length of the other side.

OPP

ADJ A

HYP

OPPOSITE .

HYPOTENUSESIN A = 11

x

SIN 37° =11

x

C

B

11 cm

37°A

x

(11) SIN 37° = x

6.62 cm = x

Therefore the length of side BC is 6.6 cm

Looking at Ratio 2

G

E

B

20

10

A A C

30

15

F

A

G

D

20

40

AB

AE 20 .

22.3

AC

AF 30 .

33.5

AD

AF 40 .

44.7

= 0.90 = 0.90= 0.90

When I look at the triangle from A ADJACENT .

HYPOTENUSE0.90

Neat Things about Ratios!!! OPP

ADJ A

HYP

22.333.5

44.7

The COSINE Ratio!

When I look at the triangle from A ADJACENT .

HYPOTENUSE0.90

There is a special name for this ratio. It is called the COSINE RATIO. We can use it to solve for A

OPP

ADJ A

HYP

ADJACENT . HYPOTENUSECOS A =

The COSINE Ratio!

If we know A, and one side, we can calculate the length of the other side.

OPP

ADJ A

HYP

ADJACENT .

HYPOTENUSECOS A = 35

x

COS 50° =35

x

C

B

35 m

50°A x

(35) COS 50° = x

22.497 m = x

Therefore the length of side AB is 22.5 m

Using the Sides to Solve for A

If we know given the side length, we can use them to solve forA

A storm caused a 13.5m lamp post to lean over. The top of the pole is now 11.5m above the ground. Find the measure of the angle between the lamp

post and the ground, to the nearest degree

11.513.5m

13.5m

Finding the Measure of an Angle

13.5 m

11.5 m

x°

4.58

851851851.0sin

851851851.0 sin5.13

5.11 sin

sin

1

x

x

x

x

Hypotenuse

Oppositex

2nd Function sin-1 11.5 13.5( ) =

Are you confused?!?!

If your finding this frustrating… don’t worry

Its been the “COS” of frustration for many math students!