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