Metric Relations in Right Triangles

• By drawing the altitude from the right angle of a right triangle, three similar right triangles are formed

• Using the lengths of the corresponding sides of the triangles formed, we can determine the ratios and from this determine certain geometric properties

Geometric Properties

Property 1

A.)

• In a right triangle the length of the leg of a right triangle is the geometric mean between the length of its projection on the hypotenuse

• In easiest terms, a leg squared is equal to the hypotenuse multiplied by the leg’s projection on the hypotenuse.

Property 2:• The square of the altitude is

equal to one part of the hypotenuse multiplied by the other

Property 3:• In a right triangle, the

product of the length of the hypotenuse and its corresponding altitude is equal to the product of the lengths of the legs.

• Visions P. 116 # 9• Visions P. 115 # 2-8, 10, 13, 15, 19• Math 3000 P. 217 # 1, 3, 4

Class Work and Homework

Trigonometry

Definition:

• Trigonometry is the art of studying triangles (in particular, but not limited to, right triangles)

• Deals with the relationships between the angles and side lengths of a triangle

• Before learning the key formulas in trigonometry, it is of absolute importance that some terms are understood

• Because we are dealing with right triangles, you are already familiar with one very important right triangle theorem:

– The Pythagorean Theorem

a² + b² = c²

• In every right triangle, because one of the angles measures 90°, then logically the other two angles must add up to 90°

A

B C

Because m<B = 90° then m<A + m<C = 90° (since there are 180° in every triangle)

Hypotenuse:– The side that is opposite the

right angle– The longest side in the right

triangleOpposite Side:

– The side that is opposite of a given angle

– Ex: Side AB is opposite m<C Side BC is opposite

m<AAdjacent Side:

– The side that is neither the hypotenuse or opposite

Ex: Side BC is adjacent to m<C Side AB is adjacent to m<A

A

B C

A

B C

Example:Fill in the side that corresponds to the following questions:

Hypotenuse: _________________

Opposite m<A: _________________

Adjacent m<A: _________________

Opposite m<C: __________________

Adjacent m<C: __________________

These three definitions of the sides are of utmost importance in trigonometry

They are at the root of finding every angle in a right triangle

The MOST important Gibberish word you will need to remember in math life

SOH – CAH - TOA

A

B C

Adjacent

OppositeA

Hypotenuse

AdjacentA

Hypotenuse

OppositeA

tan

cos

sin

Example:

A

B C

1

2

3

30°

60°

1

360tan

2

160cos

2

360sin

3

130tan

2

330cos

2

130sin

adjacent

opposite

hypotenuse

adjacent

hypotenuse

opposite

Using Your Calculator 1. The keys sin, cos, tan on the calculator

enable you to calculate the value of sin A, cos A, or tan A knowing the measure of angle A

So if you know the measure of an angle you can use the sin, cos, or tan buttons on your calculator in order to calculate its value

2. The keys sin-1, cos-1, tan-1 on the calculator enable you to calculate the measure of angle A knowing sin A

• So if know sin A, cos A, or tan A, you can calculate the measure of angle A

Homework

• Hand outs 1 and 2: Trigonometric Ratios and Calculator

• Math 3000 pages 228, 229 # 2,3,4,5P. 182 # 1P. 184 # 2

Finding Missing Sides Using

Trigonometric Ratios

Finding the Missing Side of a Right Triangle

• In order to find a missing side, you will need two pieces of information:– An angle– One side length

In a right triangle

1. Finding the measure x of side BC opposite to the known angle A, knowing also the measure of the hypotenuse, requires the use of sin A

Remember: SOH*****Cross Multiply*****

Sin 50º= x=5sin50º = 3.83 cm5

x

Finding the measure y of side AC adjacent to the known Angle A, knowing also the measure of the hypotenuse, requires the use of cos ARemember: cos = adjacent/hypotenuse

*****Cross Multiply*****

Cos 50º = y = 5 cos 50º = 3.21 cm

5

y

3. Finding the measure x of side BC opposite to the known angle A, knowing also the measure of the adjacent side to angle A, requires the use of tan A

remember tan=opposite/adjacent ***cross multiply***

tan 30º = x = 4 tan 30º = 2.31 cm4

x

Formulas for 90° triangleFormulas to find a missing side Formulas to find a missing angle

(hyp)² = (opp)² + (adj)²

( )

1

Sin A opp

hyp

( )

1

Cos A adj

hyp

( )

1

Tan A opp

adj

1sinopp

Ahyp

1cosadj

Ahyp

1tanopp

Aadj

Example

In the following triangle, find the value of side x(Remember to use SOH-CAH-TOA)!!!

sin 505

5 sin 50

3.83

x

x

x

50

x

5cm

Example

In the following triangle, find the value of side y(Remember to use SOH-CAH-TOA)!!!

50

x 5cm

cos505

5 cos50

3.21

x

x

x

Example

• Find the value of x

60

8 x

8tan 60

8

tan 604.6

x

x

x

Class work and homework

• Finding missing sides

• math 3000: page 229 # 6,7 • Hand out

Finding Missing Angles using Trigonometry Ratios

In a Right Triangle

1. Find the acute angle A when its opposite side and the hypotenuse are known requires the use of sin A

SOH – Opposite/hypotenuse

sin A = M<A=sin-1 ( )=53.1º4

5

4

5

2. Finding the acute angle A when its adjacent side and the hypotenuse are known requires the use of cos A

Cos = adjacent/hypotenuse

Cos A= m<A = cos-1 ( ) = 41.4º 3

43

4

3. Finding the acute angle A when its opposite side and adjacent side are known requires the use of tan A

tan = opposite/adjacent

Tan A = m<A=tan-1 ( ) = 56.3º3

23

2

Find the missing Angle

• If we take the inverse of each formula, we can find the missing side angle in a 90° triangle

• The symbol for the inverse of sin (A) is sin-1; cos (A) is cos-1; tan (A) is tan-1

Formulas for 90° triangleFormulas to find a missing side Formulas to find a missing angle

(hyp)² = (opp)² + (adj)²

( )

1

Sin A opp

hyp

( )

1

Cos A adj

hyp

( )

1

Tan A opp

adj

1sinopp

Ahyp

1cosadj

Ahyp

1tanopp

Aadj

Example

sin 30º = 0.5 and sin-1 (0.5) = 30º

Class work and homework

Hand out Find the missing angles using trig ratiosMath 3000 page 231 # 8,9

Math 3000 pg 232 # 10-15

10. x=10.4 cm; y=3.0 cm; z=9.0 cm; t=5.2 cm11. ab=10cos64 = 4.38 cm; bc=10sin64= 8.99 cm; area: 39.4 cm

squared12. H=50tan40 = 42 m13. Length of shadow = 50tan30 = 28.9 m14. X=65tan54 degrees = 89.5 m

15. B). c=50 degrees, ab=4,6, ac=3.9c) mc=60 decrees, ab=5.2, ac=3d) Bc=10, angle b=52.1 degrees, angle c=36.9 degreese) Ac=12, angle b=67.4, angle c=22.6f) Ab=15, angle b=28.1 degrees, angle c=61.9 degrees

Solving a triangle

To determine the measure of all its sides and angles

Sine Law

• The sides in a triangle are directly proportional to the sine of the opposite angles to these sides

sin sin sin

a b c

A B C

• The sine law can be used to find the measure of a missing side or angle

1st Case

• Finding a side when we know two angles and a side

We calculate the measure x of AC

15 15sin 5013.27

sin 50 sin 60 sin 60

xx cm

How to:

1. Place Measurement x over sin known angle2. Equal to3. Measurement known side over sin of known

angle4. Cross multiply and divide to find unknown

measurement5. Calculate.

2nd Case

• Finding an angle when we know two sides and the opposite angle to one of these two sides

• We calculate the measure of angle B

10 13 10sin 50sin 0.5893 36

sin sin 50 13B m B

B

• Make sure you have opposite angles and side measurements. Remember total inside angles must equal 180º

How to calculate if need to find an angle:

1. Place side measurement known over sin of angle we wish to know

2. Equal to 3. side measurement over sin angle we know4. Cross multiply and divide to find x5. To calculate angle –sin x = angle. Don’t forget

unit i.e.º

The sine of an obtuse angle

• The trigonometric functions (sine, cosine, etc.) are defined in a right triangle in terms of an acute angle. What, then, shall we mean by the sine of an obtuse angle ABC?

• The sine of an obtuse angle is defined to be the sine of its supplement.

• How to find the measure of the degree of an obtuse angle:

• Follow the procedure you have learned so far, then subtract that angle from 180º

18.6 cm

10 cm

22º

10 18.6

sin 22 sin

18.6sin 22

10.697

sin .697

44.2

180 44.2 135.5

inw

Class assignment

• Find all missing side lengths and angles and math 3000 page 232 # 10-15

There are three formulas to find the area of any triangle

Area of a Triangle

• The first of the three you are already aware of:

A = b x h 2

Area = ac ● sinB 2

Area = ab ● sinC 2

Area = bc ● sinA 2

A

C B

c b

a

Area of a Triangle knowing Two Sides and the Angle in Between

• These are the 3 formulas that involve the sine of an angle and the two sides that contain the angle

Example:- Find the area of the following triangle

Area = ac●sinB 2

Area = 6 ●12sin(36.3°) 2

Area = 21.3cm²

6cm

12cm

36.3°

Hero’s Formula

Where ‘p’ = HALF of the perimeter

Example:p = 6 + 8 + 12 = 26= 13 2 2

))()(( cpbpappArea

6cm8cm

12cm

)1213)(813)(613(13 Area

)1)(5)(7(13Area

23.21455 cmArea

Step 1: Label the sides of the triangle if relevantStep 2: State what information we knowStep 3: Select a theorem and find the area

• In easiest terms, you need two side lengths and the angle in between them to find the area of any triangle

Example:- Find the area of the following triangle

Area = ac●sinB 2

Area = 6 ●12sin(36.3°) 2

Area = 21.3cm²

6cm

12cm36.3°

Area of a Triangle Knowing Two Angles and One Side

Step 1: Draw the Altitude ADStep 2: Find mADStep 3: Find mBDStep 4: Find mDC

Step 5: Add mBD and mDCStep 6: Use any formula you wish to find the

total area

55 25

10cm

A

B CD

Class work and homework

• Math 3000 pg 233 act 1 and 2, and pg 235 numbers 1-5, 7 and hand outs