Embed Size (px)
Citation preview
Unit 8 – Right Triangle TrigTrigonometric Ratios
in Right Triangles
Trigonometric Ratios are based on the Concept of Similar Triangles!
All 45º- 45º- 90º Triangles are Similar!
45 º
2
2
22
45 º
1
1
2
45 º
1
2
1
2
1
All 30º- 60º- 90º Triangles are Similar!
1
60º
30º
½
23
32
60º
30º
2
4
2
60º
30º
1
3
hypotenuse
leg
leg
a
b
c
Trigonometric functions -- the ratios of sides of a right triangle.
Similar Triangles Always Have the Same Trig Ratio Answers!
SINECOSINE
TANGENT
They are abbreviated using their first 3 letters
c
a
hypotenuse
oppositesin
oppositec
b
hypotenuse
adjacentcos
adjacent
b
a
adjacent
oppositetan
This method only applies if you have a right triangle and is only for the acute angles (angles less than 90°) in the triangle.
3
45
Oh, I'm
acute!
So am I!
a
b
c
Here is a mnemonic to help you memorize the ratios.
SOHCAHTOA
c
b
hypotenuse
oppositesin
adjacentcos
hypotenuse
a
c opposite
tanadjacent
b
a
opposite
adjacent
SOHCAHTOA
It is important to note WHICH angle you are talking about when you find the value of the trig function.
a
bc
Let's try finding some trig functions with some numbers.
3
45
sin = Use a mnemonic and figure out which sides of the triangle you need for sine.
h
o5
3
opposite
hypotenuse
tan =
a
o3
4
opposite
adjacent
Use a mnemonic and figure out which sides of the triangle you need for tangent.
a
bc
How do the trig answers for and
relate to each other?
3
45opposite
hypotenuse
opposite
adjacent
Find the sine, the cosine, and the tangent of angle A.
Give a fraction and decimal answer (round to 4 places).
hyp
oppA sin
8.10
9 8333.
hyp
adjA cos
8.10
6 5556.
adj
oppA tan
6
9 5.1
9
6
10.8
A
Now, figure out your ratios.
1.9 cm
7.7 cm
14º
1.9
7.7Tangent 14º
0.25
The Tangent of an angle is the ratio of the opposite side of a triangle to its adjacent side.
oppositeadjacent
hypotenuse
3.2 cm
7.2 cm24º
3.2
7.2
Tangent 24º
0.45
Tangent A =
opposite
adjacent
As an acute angle of a triangle approaches 90º, its tangent
becomes infinitely large
Tan 89.9º = 573
Tan 89.99º = 5,730
Tangent A =
opposite
adjacent
etc.
very large
very small
Since the sine and cosine functions alwayshave the hypotenuse as the denominator,
and since the hypotenuse is the longest side,these two functions will always be less than 1.
Sine A =
opposite
hypotenuse
Cosine A =
adjacent
hypotenuse
ASine 89º = .9998
Sine 89.9º = .999998
3.2 cm7.9 cm
24º
9.7
2.3
Sin 24º
0.41
Sin α = hypotenuse
opposite
5.5 cm
7.9 cm
46º
9.7
5.5
Cos 46º
0.70
Cosine β = hypotenuse
adjacent
Ex. Solve for a missing value using a trig function.
5520 m
x
20
55tanx
m 6.28x
x55tan20tan 20 55 )
Now, figure out which trig ratio you have and set up the problem.
Ex: 2 Find the missing side. Round to the nearest tenth.
72
80 ft
x
x
8072tan
ft 26x
8072tan x
72tan
80x
tan 80 72 = ( ) )
Now, figure out which trig ratio you have and set up the problem.
Ex: 3 Find the missing side. Round to the nearest tenth.
24
283 mx 283
24sinx
m 1.115x
x24sin283
Now, figure out which trig ratio you have and set up the problem.
Ex: 4 Find the missing side. Round to the nearest tenth.
4020 ft x
2040cos
x
ft 3.15x
x40cos20
Finding an angle.(Figuring out which ratio to use and getting
to use the 2nd button and one of the trig buttons.)
Ex. 1: Find . Round to four decimal places.
9
17.2
Make sure you are in degree mode (not radians).
9
2.17tan
2nd tan 17.2 9
3789.62
)
Now, figure out which trig ratio you have and set up the problem.
Ex. 2: Find . Round to three decimal places.
23
7
Make sure you are in degree mode (not radians).
23
7cos
2nd cos 7 23
281.72
)
Ex. 3: Find . Round to three decimal places.
400
200
Make sure you are in degree mode (not radians).
400
200sin
2nd sin 200 400
30)
When we are trying to find a sidewe use sin, cos, or tan.
When we are trying to find an
angle we use sin-1, cos-1, or tan-1.
A plane takes off from an airport an an angle of 18º and a speed of 240 mph. Continuing at this speed and angle,
what is the altitude of the plane after 1 minute?
18º
x
After 60 sec., at 240 mph, the plane has traveled 4 miles
4
18º
x4
opposite
hypotenuse
SohCahToa
Sine A =
opposite
hypotenuse Sine 18 =
x
4
0.3090 =
x
4
x = 1.236 milesor
6,526 feet
1
Soh
An explorer is standing 14.3 miles from the base of Mount Everest below its highest peak. His angle of
elevation to the peak is 21º. What is the number of feet from the base of Mount Everest to its peak?
21º14.3
x
Tan 21 =
x
14.30.3839 =
x
14.3
x = 5.49 miles = 29,000 feet
1
A swimmer sees the top of a lighthouse on the edge of shore at an 18º angle. The lighthouse is
150 feet high. What is the number of feet from theswimmer to the shore?
18º
150
Tan 18 =
x
150
x
0.3249 =
150
x
0.3249x = 150
0.3249 0.3249
X = 461.7 ft1
A dragon sits atop a castle 60 feet high. An archer stands 120 feet from the point on the ground directly
below the dragon. At what angle does the archer need to aim his arrow to slay the dragon?
x
60
120
Tan x =
60
120Tan x = 0.5
Tan-1(0.5) = 26.6º
Solving a Problem withthe Tangent Ratio
60º
53 ft
h = ?
We know the angle and the We know the angle and the side adjacent to 60º. We want to side adjacent to 60º. We want to know the opposite side. Use theknow the opposite side. Use thetangent ratio:tangent ratio:
ft 92353
531
3
5360tan
h
h
h
adj
opp
1
23
Why?
A surveyor is standing 50 feet from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71.5°. How tall is the tree?
50
71.5°
?
tan 71.5°
tan 71.5° 50
y
y = 50 (tan 71.5°)
y = 50 (2.98868)
149.4y ft
Ex.
Opp
Hyp
A person is 200 yards from a river. Rather than walk directly to the river, the person walks along a straight path to the river’s edge at a 60° angle. How far must the person walk to reach the river’s edge?
200
x
Ex. 5
60°
cos 60°
x (cos 60°) = 200
x
X = 400 yards
Trigonometric Functions on a Rectangular Coordinate System
x
y
Pick a point on theterminal ray and drop a perpendicular to the x-axis.
ry
x
The adjacent side is xThe opposite side is yThe hypotenuse is labeled rThis is called a REFERENCE TRIANGLE.
y
x
x
yx
r
r
x
y
r
r
y
cottan
seccos
cscsin
Trigonometric Ratios may be found by:
45 º
1
1
2Using ratios of special trianglesUsing ratios of special triangles
145tan2
145cos
2
145sin
For angles other than 45º, 30º, 60º you will need to use a For angles other than 45º, 30º, 60º you will need to use a calculator. (Set it in Degree Mode for now.)calculator. (Set it in Degree Mode for now.)
