Essential Question?

How can we use triangles, especially right triangles, to solve problems?

Properties of Rational Exponents

Warm upInverse Variation:

Boyle’s Law states that when a sample of gas is kept at a constant temperature, the volume, V varies inversely with the pressure, P exerted on it.

•Write an equation for Boyle’s Law

•If V = 20 Liters at 500 psi, what is V if pressures is 800 psi

Trigonometric Ratios

A RATIO is a comparison of two

numbers. For example; boys to girls cats : dogs

right : wrong.

Trigonometry – study of the measurement of sides and angles in triangles

In Trigonometry, the comparison is between sides of a right triangle.

Three Trigonometric Ratios• Sine – abbreviated ‘sin’.

Ratio: sin θ = opposite side hypotenuse

Θ this is the symbol for an unknown angle measure. It’s name is ‘Theta’.

A

C Bopposite

hypotenuse

θ

Three Trigonometric Ratios

Θ this is the symbol for an unknown angle measure. It’s name is ‘Theta’.

A

C B

• Cosine - abbreviated ‘cos’.

Ratio: cos θ = adjacent side

hypotenuse

ad

jace

nt

hypotenuse

θ

Three Trigonometric Ratios

Θ this is the symbol for an unknown angle measure. It’s name is ‘Theta’.

A

C B

• Tangent - abbreviated ‘tan’.

Ratio: tan θ = opposite side adjacent

side

opposite

ad

jace

nt

θ

Easy way to remember trig ratios:

SOH CAH TOA

Three Trigonometric Ratios• Sine – abbreviated ‘sin’.

– Ratio: sin θ = opposite side

hypotenuse

• Cosine - abbreviated ‘cos’. – Ratio: cos θ = adjacent side

hypotenuse

• Tangent - abbreviated ‘tan’. – Ratio: tan θ = opposite side

adjacent side

Θ this is the symbol for an unknown angle measure. It’s name is ‘Theta’.

Trig. Ratios

Name

“say”Sine Cosine tangent

Abbreviation

Abbrev.Sin Cos Tan

Ratio of an angle measure

Sinθ = opposite side

hypotenuse

cosθ = adjacent side

hypotenuse

tanθ =opposite side

adjacent side

Make sure you have a calculator…

I want to find Use these calculator keys

sin, cos or tan

ratio

SIN

COS

TAN

Angle measure

SIN-1

COS-1

TAN-1

To set your calculator to ‘Degree’…..

•Press MODE (next to 2nd button)

•Degree (third line down… highlight it by pressing Enter

•2nd Quit Clear

Let’s practice…

B

c

a

C b A

Sin Θ =

13

12

5

OppositeHypotenuse

Cos Θ = AdjacentHypotenuse

Tan Θ = OppositeAdjacent

Sin A= Sin B =

Cos A= Cos B =

Tan A= Tan B =

Lesson 4.4 (I)Lesson 4.4 (I)

35sin6x

4.3x

25cos4x

6.3x

55tan/7x

9.4x

65sin/16x

7.17x

)4/1(tan 1x

ox 0.14

)2/1(cos 1x

ox 60

• Ex 1) How do we find the angle measure?

C A

B

Θo

18 cm

12

2nd Cos(12/18) = Cos-1(12/18)

= 48.2o

1) What is given?

2) What trig ratio?

3) What is asked for?

Find measure of <B?

Hypotenuse

Adjacent

Cos Θ = adj/hyp

Find angle Θ =

Using trig ratios in equations

Remember when you had to solve:12 = x What did you do? 6

(6) (6)

72 = x

What if x is in the denominator? 12 = 6 What did you do? x

(x) (x)

12x = 6__ __

12 12 x = 1/2

Ex 2) Let’s practice…

B

C A

Process:

1)Identify what is given

2)Which trig ratio, sin, cos, or tan will work with what is given

3)Plug in and solveX cm

40o

7.6 cm

Process:

1)Hyp = 7.6

<A = 40o and opposite = x

2) Sin = opposite/hypotenuse

3) solve:

7.6 cm

X cmSin 40o =

7.6 x Sin 40o X =

X = 4.9 cm

Ex 3) Let’s practice…

B

c A

Process:

1)Identify what is given

2)Which trig ratio, sin, cos, or tan will work with what is given

3)Plug in and solveX cm

36o 18 cm

Process:

1)Hyp = 18

<B = 36o and adjacent = x

2) Cos = adjacent/hypotenuse

3) Solve:

18 cm

X cmCos 36o =

18 x Cos 36o X =

X = 14.6 cm

• Ex 4) Let’s practice

c A

X cm

30o

18 cm

Process:

1)Hyp = x

<A = 30o and adjacent = 18

2) Cos = adjacent/hypotenuse

3) Solve:

18 cmX cmCos 30o =

Cos 30o =

Cos 30o =

18 cm

X cmX cm

Cos30o and x have to change places – Swith and divide!

X cm

X cmCos 30o = 18 cm Cos 30o

X cm = 18 cm

Cos 30o = 20.8 cm

B

Practice some more…

Ex 5) Find tan A:

C A

B

48o

5.8

x

C A

B

54o

Ex 6) What trig function would find x?

18x

Toolkit

Trig RatiosUnknown will be in one of

three places:

Sin Θ = Angle Θ

Cos Θ = Numerator:

Tan Θ = Denominator:

OppositeHypotenuse

AdjacentHypotenuse

OppositeAdjacent

2nd trig(ratio) = angle

xgivenTrig angle =

givenx

Trig angle =

Multiply

Switch and divide

Warm upGoogle

1)When and where did Pythagoras live?

2)How old is the Great Pyramid of Egypt?

3)Is it an equilateral triangle? What is the base length?

QuizDraw and label each triangle and find what is

asked for below:

1.Let side c = 15 ft. and side b = 9 ft. Find angle A and side aA

BC

c

a

b