Unit 4: Right Triangles

• Triangle Inequality

• Pythagorean Theorem and its Converse

• Trigonometry

• Inverse Trigonometry

• Solving Right Triangles

Lesson 4.1

• Triangle Inequality

• Converse of the Pythagorean Theorem

• More Classifying Triangles

Triangle Inequality

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

Practice the Triangle Inequality

• Can the following lengths represent the sides of a triangle?

1) 5, 4, 3Yes, add any two together and they are larger than the third side.

2) 5, 6, 7Yes, add any two together and they are larger than the third side.

3) 5, 5, 10No, 5+5 is equal to 10, not greater than 10.

Special Parts in a Right Triangle

• Right triangles have special names that go with it parts.

• For instance:– The two sides that form the right angle are called

the legs of the right triangle.

– The side opposite the right angle is called the hypotenuse.

• The hypotenuse is always the longest side of a right triangle.

legs

hypotenuse

Pythagorean Theorem

• c2 = a2 + b2

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

a

b

c

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

Acute Triangles fromPythagorean 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

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

Practice

Determine if the following sides create triangle. If they do determine if it is a right, obtuse, or an acute triangle.

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

Right Triangle?

Lesson 4.3

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

θ

a

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 θ

b=

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.

x7

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) = 4x =

4cos 37o=

4.7986 = 5.008

Lesson 4.4

Inverse Trigonometric Ratios: Solving for missing angles in a right

triangle.

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 used 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