Sec. 8 – 4Sec. 8 – 4Sine and Cosine Sine and Cosine

RatiosRatiosObjectives: Objectives:

1) To use sine and cosine to 1) To use sine and cosine to determine determine side lengths in side lengths in ΔΔs.s.

2) To use the2) To use the sinsin-1-1 and the cos and the cos-1-1 to solve to solve for for measures. measures.

Just Buttons on the Just Buttons on the Calculator!Calculator!

Sine and Cosine are ratios of sides in Sine and Cosine are ratios of sides in a Right a Right ΔΔ..

CosCosSiSinn

The Sine and Cosine ratio The Sine and Cosine ratio in Rt. in Rt. ΔΔss

Sin = Oppisite SideHypotenuse

Cos = Adjacent SideHypotenuse

Hyp

Opposite

Adjacent

Write the Sin and Cos Write the Sin and Cos RatioRatio

Sin T =Sin T =

Cos T = Cos T =

Sin G =Sin G =

Cos G =Cos G = T R

G

17 8

15

OppHyp

= 817

AdjHyp

= 1517

OppHyp

= 1517

AdjHyp

= 817

What do we call G and T?

Complementary

For complementary s: Sin X = Cos Y

Interesting

Ex.1: Solve for the missing Ex.1: Solve for the missing variables.variables.

20°

xx

yy

6cm

Lets solve for x first.

Sin 20 =OppHyp

Sin 20 = x6

(6)(.342) = x

2.1 = x

Now solve for y.

Cos 20 = AdjHyp

Cos 20 = y6

(6)(.939) = y

5.6 = y

The SinThe Sin-1-1 Cos Cos-1-1 and Tan and Tan-1-1

Use it when you are looking for Use it when you are looking for missing missing s.s.

CosSin

Sin-1 Cos-1

Use the shift (2nd) Key

Find the missing Find the missing ss

x°

y°

1.9

3.5

4.0

Any of the Trig ratios will work since you have all three sides.

Use the inverse sine ratio to solve for x.

Sin x =

3.54.0

Sin x = .875

Sin-1 (.875) = x

61° = x

Cos y =

3.54.0

Cos y = .875

Cos-1 (.875) = y

29° = y

Use the inverse cosine ratio to solve for y.

Pneumonic DevicePneumonic Device

S O H C A H T O A

ine pp

osite

ypote

nu

se osin

ed

jace

nt

ypote

nu

se

an

ge

nt

pp

osit

e dja

cen

t= = =

SOH CAH TOA

How do you know what trig ratio to How do you know what trig ratio to use?use?

It depends on what is given to you and It depends on what is given to you and where you are trying to go.where you are trying to go.

15

22°

x

Cosine55

40°

x

Tangent

What have you learned??What have you learned??

Sin Sin = =

Cos Cos = =

Tan Tan = =

OppositeHypotenu

seAdjacentHypotenu

se

** Use the Sin-1 and Cos-1 when looking for the measures.

OppositeAdjacent