Transcript Document

Trigonometry
Lesson 7.4
What is Trigonometry?
• Trigonometry is the study of how the sides
and angles of a triangle are related to
each other.
• It's all about triangles!
What is a Trigonometric Ratio?
• A ratio of the lengths of the sides of a right
triangle.
• The three most common trigonometric
Ratios are:
– Sine  “sin”
– Cosine  “cos”
– Tangent  “tan”
Trig Definitions: S-O-H-C-A-H-T-O-A
• Sin (angle) =
Opposite
---------------Hypotenuse
S-O-H
• Cos (angle) =
Adjacent
---------------Hypotenuse
C-A-H
• Tan (angle) =
Opposite
---------------Adjacent
T-O-A
Ways to Remember
• S-O-H
Some Old Hillbilly
Caught Another Hillbilly
Throwing Old Apples
• C-A-H
• T-O-A
She Offered Her
Cat A Heaping
Teaspoon Of Acid
Extra-credit opportunity:
Your own saying
Key Concepts of Trigonometric Ratios
A
opposite
adjacent
A
C
θ
adjacent
B
C
θ
opposite
B
What’s Constant: Side opposite right angle is the hypotenuse
What Changes: Side opposite the angle, θ; Side adjacent to the angle, θ
In the triangle to the left: AC is opposite of θ and BC is adjacent to it
In the triangle to the right: AC is adjacent to θ and BC is opposite it
Left-most Triangle:
Right-most Triangle:
opposite side
AC
sin θ = ------------------- = -----hypotenuse
AB
opposite side
BC
sin θ = ------------------- = -----hypotenuse
AB
adjacent side
BC
cos θ = ------------------- = -----hypotenuse
AB
adjacent side
AC
cos θ = ------------------- = -----hypotenuse
AB
opposite side
AC
tan θ = ------------------- = -----adjacent side
BC
opposite side
BC
tan θ = ------------------- = -----adjacent side
AC
Steps to Solve Trig Problems
• Step 1: Draw a triangle to fit problem
• Step 2: Label sides from angle’s view
– H: hypotenuse
– O: opposite
– A: adjacent
• Step 3: Identify trig function to use
– Circle what values you have or are looking for
– SOH
CAH
TOA
• Step 4: Set up equation
• Step 5: Solve for variable
Example 1
SOH-CAH-TOA (Sin θ = O/H, Cos θ = A/H, Tan θ = O/A)
θ is a symbol for an angle
Example 1: Find sin L, cos L, tan L, sin N, cos N, and tan N. Express
each ratio as a fraction and as a decimal rounded to nearest hundredth.
L
8
sin L = ------ = 0.47
17
17
15
15
cos L = ------ = 0.88
17
8
tan L = ------ = 0.53
15
N
8
M
15
sin N = ------ = 0.88
17
8
cos N = -----= 0.47
17
15
tan N = -----= 1.88
8
Example 2
SOH-CAH-TOA (Sin θ = O/H, Cos θ = A/H, Tan θ = O/A)
θ is a symbol for an angle
Example 2:
17
52°
x
When looking for a side use the appropriate trig
function (based on your angle and its
relationship to x, and your given side).
17 is opposite of the angle and x is adjacent to it:
opp, adj  use tangent
tan 52° = 17/x
x tan 52° = 17
x = 17/tan 52°
x = 13.28
Example 3
SOH-CAH-TOA (Sin θ = O/H, Cos θ = A/H, Tan θ = O/A)
θ is a symbol for an angle
Example 3:
13
x
13 is the hypotenuse (opposite from the 90
degree angle) and x is opposite from given
angle:
opp, hyp  use sin
47°
sin 47° = x/13
13 sin 47° = x
9.51 = x
Check Yourself
• You have a hypotenuse and an adjacent side
14.34
Cos
Use: _______
Solve:
x
=
___
25
55°
x
• You have an opposite and adjacent side
21.42
Tan
Use: _______
Solve: y = ___
15
35°
y
• You have an opposite side and a hypotenuse
26.15
Sin
Use: _______
Solve:
z
=
___
z
15
35°
Example 4
EXERCISING A fitness trainer sets the incline on a
treadmill to 7°. The walking surface is 5 feet long.
Approximately how many inches did the trainer raise the
end of the treadmill from the floor?
Step 1:
Step 2:
Step 3:
Step 4:
Draw a triangle to fit problem
Label sides from angle’s view
Identify trig function to use
Set up equation
Step 5: Solve for variable
Opp
SO/H
CA/H
TO/A
Opp
y
sin 7° = -------- = ----Hyp
60
Let y be the height of the treadmill from the floor in inches.
The length of the treadmill is 5 feet, or 60 inches.
Example 4 cont
Multiply each side by 60.
Use a calculator to find y.
KEYSTROKES: 60
SIN
7
ENTER
7.312160604
Answer: The treadmill is about 7.3 inches high.
Example 5
CONSTRUCTION The bottom of a handicap ramp is
15 feet from the entrance of a building. If the angle of
the ramp is about 4.8°, how high does the ramp rise
off the ground to the nearest inch?
Answer: about 15 in.
How to Find an Angle
• In trigonometry, you can find the measure
of the angle by using the inverse of sine,
cosine, and tangent.
• Sin (angle) = x  angle = sin-1 (x)
• Cos (angle) = y
 angle = cos-1 (y)
• Tan (angle) = z
 angle = tan-1 (z)
Example 6
SOH-CAH-TOA (Sin θ = O/H, Cos θ = A/H, Tan θ = O/A)
θ is a symbol for an angle
Example 6:
When looking for an angle use the
inverse of the appropriate trig function
(2nd key then trig function on your
calculator)
12
x°
8
tan x° = 12/8
x = tan-1 (12/8)
x = 56.31°
12 is opposite the angle x; and 8 is
adjacent to it:
opp, adj  use tangent
Trig Practice
1) Identify what you are trying to find (variable) – Side or Angle
2) Relate given (opp, adj, hyp, angle) to the variable
3) Solve for variable
1.
Side, x opposite 30° and 24 is the hyp
 sin 30 = x/24
24
x
x = 24 sin 30 = 12
30°
2.
20
15
Angle, x opposite 20 leg and 15 is adj leg
 tan x = 20/15 x = tan-1 (20/15) = 53.1
x°
3.
x
60°
30
Side, x adjacent 60 and 30 is the hyp
 cos 60 = x/30 x = 30 cos 60 = 15
Trig Practice cont
1) Identify what you are trying to find (variable) – Side or Angle
2) Relate given (opp, adj, hyp, angle) to the variable
3) Solve for variable
4.
13
x
Side, x is hypotenuse and 12 is adj leg 
cos 45 = 12/x x = 12/(cos 45) = 12√2
49°
12
5.
45°
x
6.
x°
12
18
Side, x opposite 49 and 13 is the hyp
 sin 49 = x/13 x = 13 sin 49 = 9.81
Angle, x is opposite 12 and 18 is hyp  sin
x = 12/18 x = cos -1 (12/18) = 48.2
Trig Practice cont
1) Identify what you are trying to find (variable) – Side or Angle
2) Relate given (opp, adj, hyp, angle) to the variable
3) Solve for variable
7.
16
Side, x is adjacent 54 and 16 is opp
 tan 54 = 16/x x = 16/(tan 54) = 11.62
54°
x
8.
Angle, x is opposite 12 and adj to 10
 tan x = 12/10 x = tan-1 (12/10) = 50.2
12
x°
10
Summary & Homework
• Summary:
– Trigonometric ratios can be used to find measures
in right triangles
– Identify what you are trying to find (variable) – Side
or Angle
– Relate given (opp, adj, hyp, angle) to the variable
– Solve for variable
• Homework:
– Pg. 368 # 19-51 odd (15 problems)