Chapter 9

Right Trianglesand

Trigonometry

Chapter 9 Objectives

• Prove Pythagorean Theorem• Utilize Pythagorean converse• Identify right, obtuse, and acute triangles

using Pythagorean converse• Define properties of a 45-45-90 triangle• Define properties of a 30-60-90 triangle• Apply trigonometry to geometric

situations• Utilize inverse trigonometry functions

Lesson 9.2

The Pythagorean Theorem

Theorem 9.4:Pythagorean Theorem

• In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

• c2 = a2 + b2

– c is always the hypotenuse– a and b are the legs in any order

a

b

c

Proof of Pythagorean Theorem

a

a

a

a

b

b

b

b

c

c c

c

How would you find the area of the figure at right?

(Hint: 2 ways)

(a + b)2

4(1/2ab) + c2

Side length squared

Area of triangles plus area of inner square

(a + b)(a + b)

a2 + 2ab + b2 2ab + c2

Should give same answer

=-2ab -2ab

a2 + b2 = c2

Example 1

Find the length of the missing sideI. a = 3, b = 3

I. c2 = a2 + b2

c2 = 32 + 32

c2 = 9 + 9c2 = 18c = 18 c = 32 4.2

a = 5, c = 13II. c2 = a2 + b2

132 = 52 + b2

169 = 25 + b2

b2 = 169 - 25b2 = 144b = 144b = 12

c2 = a2 + b2

Pythagorean Triple

• A Pythagorean triple is a set of three positive integers that satisfy the Pythagorean Theorem.– That means that a,b, and c will all

be nice whole numbers• except 0

• 3,4,5• 5,12,13

Example 2

Find the length of the missing side and determine if the sides form a Pythagorean Triple

I. a = 18, b = 24I. c2 = a2 + b2

c2 = 182 + 242

c2 = 324 + 576c2 = 900c = 900 c = 30Yes, a Pythagorean Triple is formed!

c2 = a2 + b2

Lesson 9.3

Converse of thePythagorean Theorem

Theorem 5.13:Triangle Inequality

• The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Add each combination of two sides to make sure that they are longer than the third remaining side.

6 6 6

133 2 4 4

Theorem 9.5:Converse of the

Pythagorean Theorem• If c2 = a2 + b2 is true, then the

triangle in question is a right triangle.– You need to verify the three sides of the

triangle given will make the Pythagorean Theorem true when plugged in.

– Remember the largest number given is always the hypotenuse

• Which is c in the Pythagorean Theorem

Example 3Determine whether the following

triangles are right triangles or not.

I. 4,7,10 I. c2 = a2 + b2

102 = 42 + 72

100 = 16 + 49100 65NO

II. 5,12,13I. c2 = a2 + b2

132 = 52 + 122

169 = 25 + 144169 = 169YES

Theorem 9.6:Acute Triangles from

Pythagorean Theorem• If c2 < a2 + b2, then the triangle is an acute

triangle.– So when you check if it is a right triangle and

the answer for c2 is smaller than the answer for a2 + b2, then the triangle must be acute

• It essentially means the hypotenuse shrunk a little!

• And the only way to make it shrink is to make the right angle shrink as well!

a

b

c

Theorem 9.7:Obtuse Triangles fromPythagorean Theorem

• If c2 > a2 + b2, then the triangle is an obtuse triangle.– So when you check if it is a right triangle and

the answer for c2 is larger than the answer for a2 + b2, then the triangle must be obtuse

• It essentially means the hypotenuse grew a little!• And the only way to make it grow is to make the

right angle grow as well!

a

b

c

Example 4

Determine if the following sides create a right, obtuse, acute, or no triangle at all.

A) 38, 77, 86 B) 10.5, 36.5, 37.5

c2 = a2 + b2

clongest side

c2 = a2 + b2

862 = 382 + 772 37.52 = 10.52 + 36.52

7396 = 1444 + 5929 1406.25 = 110.25 + 1332.25

7396 = 7373 1406.25 = 1442.57396 > 7373 1406.25 < 1442.5

obtuse acute

Triangle Y/N Triangle Y/N

Lesson 9.5

Trigonometric Ratios

Trigonometric Ratios

• A trigonometric ratio is a ratio of the lengths of any two sides in a right triangle.

• You must know:– one angle in the triangle other than the

right angle– one side (any side) of the triangle.

• These help find any other side of the triangle.

Sine• The sine is a ratio of

– side opposite the known angle, and…– the hypotenuse

• Abbreviated– sin

• This is used to find one of those sides.– Use your known angle as a reference point

θa

b

c

sin θ = side opposite θ

hypotenuse

b= c

Cosine• The cosine is a ratio of

– side adjacent the known angle, and…– the hypotenuse

• Abbreviated– cos

• This is used to find one of those sides.– Use your known angle as a reference point

θa

b

c

cos θ = side adjacent θ

hypotenuse

a= c

Tangent• The tangent is a ratio of

– side opposite the known angle, and…

– side adjacent the known angle• Abbreviated

– tan• This is used to find one of those sides.

– Use your known angle as a reference point

θa

b

c

tan θ = side opposite θ

side adjacent θb

= a

SOHCAHTOA• This is a handy way

of remembering which ratio involves which components.

• Remember to start at the known angle as the reference point.

• Also, each combination is a ratio– So the sin is the

opposite side divided by the hypotenuse

Soh

Cah

Toa

in

pposite

ypotenuse

os

djacent

ypotenuse

anpposite

djacent

Example 5

• First determine which trig function you want to use by identifying the known parts and the variable side.

• Use that function on your calculator to find the decimal equivalent for the angle.

• Set that number equal to the ratio of side lengths and solve for the variable side using algebra.

If you do not have a calculator with trig buttons, then turn to p845 in book for a table of all trig ratios up to 90o. x

7

42o

4x37o

xsin 42o =

77 (sin 42o) = x

7 (.6691) = x = 4.683

4cos 37o = x

Get x out of denominator first by multiplying both sides by x.

x (cos 37o) = 4

x =4

cos 37o =4

.7986= 5.008

Lesson 9.4

Special Right Triangles

Theorem 9.8:45-45-90 Triangle

• In a 45-45-90 triangle, the hypotenuse is 2 times as long as each leg.– Remember each leg is the same length

• Theorem 4.7: Converse of the Base Angles

1

1

245o

45o

Example 6• If you are needing to

find the length of the hypotenuse, simply multiply the length of either leg by 2.– Best way to do that

is simply write the leg followed by 2.

• When you know the hypotenuse, find the number in front of the 2 and that is the length of your leg.

• If there is no 2, then you need to divide by 2 on your calculator.

45o

45o

4

4

4 2 45o

45o

6

6

6 2

I II

Find the length of the missing side(s)

Theorem 9.9:30-60-90 Triangle

• In a 30-60-90 triangle– the hypotenuse is twice the length of the

shortest leg, and…– the longer leg is the 3 times as long as

the shortest leg.

12

3

60o

30o

Example 7

• Knowing the shortest leg and trying to find the hypotenuse, simply multiply the shortest leg by 2.

• If you need to find the shortest leg, then divide the hypotenuse by 2.

• Knowing the shortest leg and trying to find the longer leg, multiply the shortest leg by 3.– Or just write 3

after the length of the shortest leg.

48

4 3

60o

30o

Find the length of the missing side(s)

Lesson 9.6

Solving Right Triangles

Inverse Trig RatiosInverse trig ratios are used to find the measure of the angles of a triangle.

The catch is…you must know two side lengths.

Those sides determine which ratio to use based on the same ratios we had from before.

SOHCAHTOA

Finding Side Lengths

Finding Angle Measures

sin sin-1

cos cos-1

tan tan-1

Example 8

• You still base your ratio on what sides are you working with compared to the angle you want to find.

• Only now, your variable is θ.• So once you find your ratio, you will then use the

inverse function of your ratio from your calculator

47

θ

917θ

SOHCAHTOA

sin θ = 4

7θ = sin-1

θ = sin-1 .5714

θ = 34.8o

cos θ = 917

θ = cos-1

θ = cos-1 .5294

4

7917

θ = 58.0o

Solving a Triangle

• To solve a right triangle, you must find– all 3 sides– all 3 angles

• or the other 2 angles besides the right angle

• So your final answer when solving a right triangle will have six parts to the answer!

Lesson 9.7

Vectors

Vector Magnitude

• The magnitude of a vector, AB, is the distance from the initial point A to the terminal point B.– It is written as AB

• Because the magnitude of a vector is essentially finding the distance between two points, we can use the Distance Formula to calculate the magnitude of the vector. AB = [(x2 – x1)2 + (y2 – y1)2]

• Or if you know the component form of the vector, you can use the Pythagorean Theorem to calculate the magnitude.

• x2 + y2 = AB2

Example 9

Calculate the magnitude of the vector AB.I. A(0,0) , B(4,5)

I. AB = [(x2 – x1)2 + (y2 – y1)2]

AB = [(4 – 0)2 + (5 – 0)2]AB = [42 + 52]AB = [(16 + 25]AB = 41 6.4

• AB = <-4,-2>II. x2 + y2 = AB2

(-4)2 + (-2)2 =AB2

16 + 4 =AB2

AB2 = 20AB = 20 4.5

Direction of a Vector• The direction of a vector, , is determined by

the angle it makes with a horizontal line that intersects the initial point A.

• Since the vector has a horizontal and vertical component, we can fill in the triangle to use a trigonometric ratio in calculating the direction of a vector.

• Knowing the components of the vector leave us with the inverse tangent to calculate the necessary angle = tan-1 y/x

• Two vectors are equal if they have the same magnitude and direction.

– However, they need not share the same initial or terminal points.

• Two vectors are parallel if they have the same or opposite direction.

– The opposite direction would be 180o different than the original.

Adding Vectors

• The sum of any two vectors is the individual sums of the horizontal and vertical components.u = <a,b>v = <c,d>u+ v = <a+c,b+d>