Transcript Document

Transparency 7-5a
5-Minute Check on Lesson 7-4a
Find x.
x ≈ 53.14°
1.
10
x°
x ≈ 17.43
32
2.
x
6
33°
3. Given an adjacent side and the hypotenuse, which trig function do
you use?
cos
4. Given an opposite side and the hypotenuse, which trig function do
you use?
sin
5. ∆MNP is a 45°- 45°- 90° triangle with right angle P. Find NP if MN =
20.
10√2 ≈ 14.14
6.
In the right triangle which
trig function would you use to find CD with C?
Standardized Test Practice:
A
cos
B
sin
C
tan
Click the mouse button or press the
Space Bar to display the answers.
C
D
37°
5
D
sec
E
Lesson 7-4b
Right Triangle Trigonometry
Trig Definitions
• 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
Some Old Hippie
Caught Another Hippie
Tripping On Acid
Extra-credit:
Your saying
Crash Trig Course
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
sin2 θ + cos2 θ = 1 (from Pythagorean Theorem)
Remember:
Sin 90° is 1
Cos 90° is 0
Tan 90° is undefined
Example 1:
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
Example 2
SOH-CAH-TOA (Sin θ = O/H, Cos θ = A/H, Tan θ = O/A)
θ is a symbol for an angle
Remember:
Sin 90° is 1
Cos 90° is 0
Tan 90° is undefined
Example 2:
17
52°
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
x
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
Remember:
Sin 90° is 1
Cos 90° is 0
Tan 90° is undefined
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
x
55°
• 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: Draw a triangle to fit problem
Step 2: Label sides from angle’s view
Step 3: Identify trig function to use
Step 4: 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.
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, 18-21, 43-46, 61