Chapter 7

Right Triangles and Trigonometry

7.1 Geometric Mean

Means

In the past, we have thought of Means as the average. That is, the mean between 2 and 10 is 6 b/c (2+10)/2 = 6

You could notice that if you take 2 (starting value) and add 4 you will get 6 (the mean).

If you take 6 (the mean) and add 4 again, you get 10 (ending value)

Because you have added the same number twice, we call this the Arithmetic Mean.

Today we will learn about Geometric Means.

Geometric Means

If for the Arithmetic Mean you add the same number each time, what do you think you do to get the geometric mean?

Right, you multiply by the same number each time.

Let s be the starting number, e be the ending number and g be the geometric mean.

So, sx = g and gx = e.Solve both for x you get x=g/s and x=e/g

Continue

Since both equations equal the same thing,you can combine them.

e gg s

Notice that the geometric means are written twice on one diagonal!Notice that the starting number and the ending number are on the other diagonal!

The pattern will always be the same, the GM’s are on one diagonal.

Find the GM between…

Find the Geometric Mean between 4 and 16.

Set up the equation 4/GM = GM/16Cross Multiply and you get GM2 = 64Taking the square root of each side you

get GM = + 8 but only +8 is between 4 and 16, then the GM between 4 and 16 is 8!

So, Why is Geometric Mean so important?

Geometric Means

AB BC ACAD DB AB

A

C

B

DThese three triangles are similar!

ΔABC ~ ΔADB ~ ΔBDC

1

2

34

AC ABAB AD

Rearranging the proportions:

Notice AB is written twice on the diagonal?So, AB is the GM between AD and AC!

Geometric Means Con’t

AB BC ACBD DC BC

A

C

B

ΔABC ~ ΔADB ~ ΔBDC

AC BCBC DC

Rearranging the proportions:

Notice BC is written twice on the diagonal?So, BC is the GM between DC and AC!

D

Geometric Mean Con’t

AD DB ABBD DC BC

A

C

B

ΔABC ~ ΔADB ~ ΔBDC

AD DBBD DC

Rearranging the proportions:

Notice BD is written twice on the diagonal?So, BD is the GM between DC and AD!

D

7.2 Pythagorean Theorem and Converse

Pythagorean Theorem

Pythagoras recognized a very important relationship between the sides (legs) and the hypotenuse of a right triangle.

That is “The sum of the squares of the legs equals the square of the hypotenuse.”

A

B

C

a

b

cHere we have a2 + b2 = c2

I like leg2 + leg2 = Hypot2

Converse of Pythagorean Theorem

Did you notice that the only triangle that uses the Pythagorean theorem is a right triangle?

So, if the “sum of the squares of the two smaller sides equals the square of the largest side” then the triangle is a Right Triangle.

If leg2 + leg2 = Hypot2, then the triangle is a right triangle.

Corollaries of Pythagoras

There are two corollaries of the Pythagorean theorem that allows you to classify a triangle by angles.

If a2 + b2 > c2 (where a and b are the two smallest sides), then the triangle is acute.

If a2 + b2 < c2 (where a and b are the two smallest sides), then the triangle is obtuse.

Pythagorean Triple

There are all sorts of side combinations that we can use to make a right triangle but some are more special.

When all three sides of a right triangle are integers, then these three numbers are called a “Pythagorean Triple”

Some common triples are 3, 4, 5 or 5, 12, 13, or 7, 24, 25…..

Plus all families of these… 3x, 4x, 5x etc..

7.3 Special Right Triangles

Special Right Triangles

Let’s take this equilateral triangle, all three sides and angles are congruent and all three angles measure 60°

60°

60° 60°Now let us drop an altitude from the top of the triangle to the horizontal side. Remember that when an altitude is drawn from the vertex angle of an “isosceles” triangle it is also an angle bisector and a median?

Special Right Triangle (Con’t)

60°

60° 60°

So, what do we know about the bottom side and what do we know about what is happening at the top angle?

The bottom segment is divided in ½ and so is the vertex angle.

If the side is s then each piece is ½ s.Each angle is 30°

30°-60°-90° Right Triangles

60°

30°

Short Leg

Long

Leg

Hypotenuse

Short Leg (SL) – This is the leg opposite the 30° angle.

Long Leg (LL) – This is the leg opposite the 60° angle.

Hypotenuse (H) – This is the side opposite the right angle.Rules:SL → H, SL x 2 = HSL → LL, SL x √3 = LL

H → SL, H/ 2 = SL LL → SL, LL / √3 = SL

45°- 45°- 90° Right Triangles

Let us look at this square, it is equilateral and equiangular.Let us divide the square by drawing a diagonal.

Now we have an “Isosceles, Right Triangle” or a 45-45-90 Right Triangle.If we make the legs 1, then what is the length of the Hypotenuse? √2

45°- 45°- 90° Right Triangles

Rules Recap:45 – 45 - 90

Leg to Hypotenuse: L x √2 = HHypotenuse to Leg: H ÷ √2 = L

30 – 60 – 90SL to Hypotenuse: SL x 2 = HHypotenuse to SL: H ÷ 2 = SLSL to LL: SL x √3 = LLLL to SL: LL ÷ √3 = SL

Practice

A C

B

a

b

c

ab c

SL LL Hypot2

63

3√35

2√22√2

30°

Practice

A C

B

a

b

c

ab c

SL LL Hypot2 2√3 43 3√3 63 3√3 63 3√3 6

5/2 5√3/2 52√2 2√6 4√2√2 √6 2√2

More Practice

A B

C

a

c

b Leg Leg Hypot3

46√27√28

3√24√3

45°

ac b

More Practice

A B

C

a

c

b Leg Leg Hypot3 3 3√24 4 4√26 6 6√27 7 7√24√2 4√2 83√2 3√2 64√3 4√3 4√6

45°

ac b

7.4 Trigonometry

Why Trig?

Up to now we have been able to find sides and or angles of certain types of Right Triangles.

If we knew two sides, we could use the Pythagorean Theorem to find the third side.

If we saw it followed a pattern, say a Pythagorean Triple, we could find missing sides.

If it was one of the two special right triangles, we could apply the rules.

What happens if we can’t fit any of these situations?

Why Trig?

We can use Right Triangle trig if we have a right triangle and a known acute angle and side to find all the missing sides.

We can use Right Triangle trig if we have a right triangle and we know two sides to find any missing acute angle.

Trig Ratios

There are three important trig ratios we will use in geometry (there actually are six – but that will have to wait 2 years).

They are Sine (Sin), Cosine (Cos) and Tangent (Tan).

You can see the Sin, Cos and Tan buttons on your calculator.

Before we do anything, make sure you are in Degree Mode vice Radian Mode.

Trig Ratios (Con’t)

The sine of an acute angle is the ratio of the opposite side over the hypotenuse.

The cosine of an acute angle is the ratio of the adjacent side over the hypotenuse.

The tangent of an acute angle is the ratio of the opposite side over the adjacent side.

SOH/CAH/TOASome Old Hippie, Caught Another Hippie,

Tripp’n Over Animals.

Sine and Cosine

A B

C

a

c

bSin <A = Opposite Side Hypotenuse

= a b

Sin <C = Opposite Side Hypotenuse

= c b

Cos <A = Adjacent Side Hypotenuse

= c b

cos <C = Adjacent Side Hypotenuse

= a b

Tangent

A B

C

a

c

bTan <A = Opposite Side Adjacent Side

= a c

Tan <C = Opposite Side Adjacent Side

= c a

To find the missing sides of a right triangle, you must use either the Sine, Cosine or Tangent functions Depending on what they give you in the problem.

Calculator

Turn your calculator on… got to Mode, make sure you’re on Degree Mode.

To find the sine of 39° all you need to do is type “sin 39” and you’ll get …

sin 39° = .629320391What does that mean?That is the ratio of the length of the

opposite side of the 39° angle over the length of the hypotenuse.

Calculator (Con’t)

Find the cosine of 47°cos 47° = .6819983601That is the ratio of the adjacent side over

the hypotenuse is .6819983601Find the tangent of 21°tan 21° = .383864035That is the ratio of the opposite side over

the adjacent side.

Example

A B

C

a

c

b = 20

25°

Given this triangle find a.

What do we know?

We know we have a right triangle, an acute angle that measures 25° and the length of the hypotenuse.We want to find a That is the opposite side,So which function?

Example (Con’t)

A B

C

a

c

b = 20

25°

We have or want the opposite side and the hypotenuse… what function has Opp Side and Hypot?

Sine function…. So sin 25° = a/20.

Solving for a …. a = 20 sin 25° or approx 8.45Now lets find c …. cos 25° = c/20Solving for c …. c = 20 cos 25° or approx 18.13

Another Example

B

C

a

C = 20

b

35°

Given this triangle find a.

What do we know?

We know we have a right triangle, an acute angle that measures 35° and the length of the adjacent side from <A.We want to find a That is the opposite side,So which function?

Example (Con’t)

A B

C

a

c = 20

b

35°

We have or want the opposite side and the adjacent side… what function has Opp Side and Adj Side?

Tangent function…. So tan 35° = a/20.

Solving for a …. a = 20 tan 35° or approx 14.00Now lets find b …. cos 35° = 20/bSolving for c …. b = 20/cos 35° or approx 24.42

Angles?

So, all the examples we have done so far have had you find sides. You used sine, cosine and tangent functions.

To find angles, it is pretty similar except you will use Inverse Sine, Inverse Cosine and Inverse Tangent.

Inverse Sine is sin -1.Inverse Cosine is cos -1.Inverse Tangent is tan -1.

Angles (Con’t)

So to find the angle that has a sine of .8543 all you need to do is type sin -1 (.8543) and you’ll get the angle measurement.

sin -1 (.8543) = 58.7°

Example

A B

Ca = 15

c = 20

bFind the measure of <A

What do we know?

We know from Angle A , that a = 15 and c = 20.a is the opposite side, c is the adjacent sidefrom <A so we’re going to use Inv TanSo, tan -1 (15/20) = 36.9°

7.5 Angles of Elevation and Depression

Angle of Elevation

How do you go about findingthe height of a building that a person is standing on?You can use right triangle trig.So you need to draw a right triangle.

Angle of Elevation and Depression

Horizontal

Angle of Elevation

Angle of Depression

Horizontal

< of Elevation is measured from the horizontal up to the “line of sight.”

< of Depression is measured from the horizontal down to the “line of sight.”

Angle of Elevation and Depression

The measurements of the angle of elevation and depression are the same because they are….

Alternate Interior Angles made by a transversal “line of sight” cutting parallel lines “horizontal lines.”

Do not think that the complementary angle of the angle of elevation is the angle of depression.

Example

Find the height of a cliff if you are in a sail boat 2000’ from the cliff and the angle of elevation to the top is 13°

tan132000y

13°2000’

y 2000 tan13y

461.74 'y

7.6 Law of Sines (H)

What if?

What if you don’t have a right triangle?Then depending on the information you

can use either the Law of Sines or the Law of Cosines.

We will only cover the Law of Sines in this class, we will leave the Law of Cosines for Pre Calculus.

Law of Sines

A

h

b

c

a

C

BDΔACDsin A = h/bh = b sin A

ΔBCDsin B = h/ah = a sin B

b sin A = a sin B

Law of Sines

sin sin sinA B Ca b c

A

b = 10

c

a

C

B45°35°

Example

sin sin sinA B Ca b c

sin35 sin 45 sin10010a c

10sin 35 sin 45a

A

10

c

a

C

B45°35°

10sin35sin 45

a

8.11a

sin 45 10sin100c 10sin100sin 45

c

13.93c

100°

Unit Circle and Radians

Unit Circle

Unit Circle – a circlew/ a radius = 1.

A radian – an anglemeasurement that gives an arc length = to the radius.

r = 1

Arc Length = 1

< meas = 1.

Unit Circle

30° or π/6

45° or π/460° or π/3

90° or π/2

315° or 7π/4

120° or 2π/3

135° or 3π/4

150° or 5π/6

180° or π

210° or 7π/6

225° or 5π/4

240° or 4π/3 270° or 3π/2

330° or 11π/6

300° or 5π/3

0°/360° or 2π

Radians

Radians is a unit that we use to measure angles. It is different unit of measure than degrees. Just like cm’s measure length and so do inches.

Conversion factor πr = 180°The “r” is to show you that this is a radian

measure (not the ratio of Circumference divided by the diameter)

Convert degrees to radians mult by (π/180°)Convert radians to degrees mult by (180°/π)

Examples:

Given 3π/4 convert to degrees.Take (3π/4)(180°/π)You get 135°Given 225° convert to π form radians.Take (225°)(π/180°)You get 5π/4