Old stuff will be used in this section› Triangle Sum Theorem

The sum of the measures of the angles in a triangle is 180°

› Pythagorean Theorem In a right triangle with legs a and b and

hypotenuse c, a2 + b2 = c2

Finding a side of a Triangle› Find side x in the right triangle below

In this figure, we’re given:An angle (65°)The Hypotenuse (8)A side Adjacent to 65°

(x)The sides we’re using are A

and Husing SOH-CAH-TOA

means we use the cosine function

65°

8

x

8cos 8 865

3.3809

x

x

Finding an Angle of a Triangle› Find the measure of the angle θ in the triangle below

In this triangle, we’re given all three

side lengths, so we can use any of

the trigonometric ratios to solve.

› SOH sin θ = 3/5 → sin-1(3/5) = 36.8699°

› CAH cos θ = 4/5 → cos-1(4/5) = 36.8699°

› TOA tan θ = 3/4 → tan-1(3/4) = 36.8699°

› All ratios give us the same answer: 36.8699°

θ

4

5

3

Solving a Right Triangle› Solve the right triangle below

The Triangle Sum Theorem helps find θ

75° + θ + 90° = 180°θ = 15°

We can use the hypotenuse (17) and the

75° angle to find sides a and b75°

17

b

a

θ

17

17

sin 75

1

17 17

17 17

6.42

cos75

4.40

a

b

a

b

Solving a Right Triangle› Solve the right triangle below

The Pythagorean Theorem helps find aa2 + 62 = 122

a2 = 108a =

We can find β by using the cosine function

cos β = 6/12cos β = 1/2cos-1(1/2) = β60° = β

We can either find θ by using the sin function

or by using The Triangle Sum Theorem θ = 30°

β

12

6

a

θ

108 6 3

Assignment› Page 429

Problems 1 – 35, odd problems Questions where you’re told to not use a

calculator can be solved using the chart you copied yesterday.

Applications› A straight road leads from an ocean beach at a constant

upward angle of 3°. How high above sea level is the road at a point 1 mile from the beach? Answer

If one is not drawn, DRAW A DIAGRAM.

Looking for the side on the right of the triangle, which is the side opposite of 3°

sin 3° = x/52805280 • sin 3° = x276.33 ft = x

Applications› According to the safety sticker on a 20-foot ladder, the

distance from the bottom of the ladder to the base of the wall on which it leans should be one-fourth of the length of the ladder: 5 feet. How high up the wall will the ladder reach If the ladder is in this position, what angle does it make with

the ground?

Draw a diagram

Applications› The wall height can be found

using the Pythagorean Theorem 52 + h2 = 202

h2 = 400 – 25 h2 = 375 h = (375)½ ≈ 19.36 ft

› We’re given the side adjacent to θ and the hypotenuse, meaning we need to use cosine cos θ = 5/20 θ = cos-1(5/20) ≈ 75.5°

Angles of Elevation and Depression

› Both create right angles from an endpoint. Angles of elevation look up; angles of depression look down.

Angle of elevation

Angle of depression

Elevation/Depression› A flagpole casts a 60-foot shadow when the angle of

elevation of the sun is 35°. Find the height of the flagpole. Draw a diagram You’re given a 35° angle

You’re given the side adjacent You’re looking for the side opposite

You’re using tangent

tan 35° = x / 60 60 • tan 35° = x 42.012 ≈ x

Elevation/Depression (#3: both)› A person on the edge of a canal observes a lamp post

on the other side with an angle of elevation of 12° to the top of the lamp post and an angle of depression of 7° to the bottom of the lamp post from eye level. The person’s eye level is 152 cm. Find the width of the canal. Find the height of the lamp post. Draw a diagram

12°

7°152 cm

152 cm

canal

top half of lamp post

Elevation/Depression (#3: both)› The canal is adjacent

to the 7° angle› You’re given 152 cm,

which is opposite 7° Use tangent tan 7° = 152 / x x = 152 / tan 7° x = 1237.94 cm

› Use the canal measurement to find the top half of the lamp post… again using tangent. tan 12° = y / 1237.94 1237.94 • tan 12° = y 263.13 cm = y

› So the height of the lamp post is 263.13 + 152 = 415.13 cm

12°

7°152 cm

canal (x)

top half of lamp post (y)

Assignment› Page 431

Problems 37 – 49, odd problems

› Quiz tomorrow1)DMS/decimal conversion2)Finding 6 trig ratios3)Solving right triangles4)A word problem or two