Transcript 5.4 Trig. Ratios

5.4 Trig. Ratios
Trigonometer
Trigon- Greek for triangles
Metric- Greek for Measure
Trig Ratios
Trig. Ratios are used to find……
• A missing side on a right triangle using one
given side and an angle.
Ex:
3cm
30o
Trig Ratios Continued …
Trig Ratios are also used to find…
• A missing angle on a right triangle using
two given two sides.
Ex.
3cm

4cm
Identifying sides of a triangle
• Sides of a triangle are referenced to the angle
Adjacent

Opposite
Hypotenuse
What if the angle is in the other
corner?
• Sides are identified according to the reference
angle.
Opposite
Hypotenuse

Adjacent
Trig Ratios Continued…
• Trig Ratios are constant values of right
triangles that are based on the ratios of
side measurements.
SOH CAH TOA
OPPOSITE
SIN 
HYPOTENUSE
ADJACENT
COS 
HYPOTENUSE
OPPOSITE
TAN 
ADJACENT
Trig Ratios
5cm
13cm

• If you want to find the
angle in degrees you
can use your
calculator or a chart
12cm
opp 5
sin( ) 
  .3846
hyp 13
adj 12
cos( ) 

 .9231
hyp 13
opp 5
tan( ) 

 .4167
adj 12
Finding the Angle in Degrees
• We already found this
information
opp 5
sin( ) 
  .3846
hyp 13
adj 12
cos( ) 

 .9231
hyp 13
opp 5
tan( ) 

 .4167
adj 12
• Make sure your calculator
is in degrees.
• Punch in 2nd Function
sin(.3846)
• Your answer should be
22.62 degrees
• Punch in 2nd Function
cos(.9231) and your
answer should still be
22.62 degrees
• And the same thing if you
do it for Tan
Finding the angle given 2 sides of a
triangle
8cm
12cm

What are we going to
use Sin, Cos or Tan?
• We have to use Sin
WHY?
• We have to use sin
because we have the
opposite side and the
hypotenuse
8
sin( )   .6667
12
sin 1  (sin( )  sin 1 .6667
  41.81o
Finding the angles of a triangle
• 6
15
• What ratio are we
going to use sin cos
or tan?
opp
hyp
adj
cos( ) 
hyp
opp
tan( ) 
adj
sin( ) 
• We have to use cos
because we have the
adjacent side and the
hypotenuse side.
adj 6
cos( ) 
  .4
hyp 15
cos cos( )  cos (.4)
1
  23.58
1
o
Finding two angles given 2 sides


15
30
What do we use sin,
cos or tan when
looking for the
angles?
opp 30
tan  

2
adj 15
tan 1 tan(  )  tan 1 (2)
  63.43
15
tan( ) 
 .5
30
tan 1 tan( )  tan 1 (.5)
  26.56
Find the missing Angle
9
15
8

opp
sin( ) 
hyp
adj
cos( ) 
hyp
opp
tan( ) 
adj

20
Solutions
Example #1
9
sin( )   .6
15
sin 1 sin( )  sin 1 (.6)
  36.87 o
Example #2
8
tan( ) 
 .4
20
1
1
tan tan( )  tan (.4)
  21.8
o
Trig Ratios Part 2
• Using Ratios to
find the missing
side.
y
Lets use
opp
Tan 
hyp
y
Tan(45 ) 
15
o
Now solve for y by multiplying both
sides by 15
X
45 o
15
You have to decide what formula you
are going to use Sin, Cos or Tan
y
Tan(45)(15)  (15)
15
Tan(45) *15  x
15  x
Trig Ratios Part 2
You have to decide what formula you are
going to use to solve for x Sin, Cos or Tan
• Using Ratios to
find the missing
side.
y
Lets use cos
15
cos( 45) 
We have to move the x to the other
x side if we
want to solve for X so to do this we must
multiply both sides by x.
15
cos( 45) * X  * X
X
X
45
15
o
The x’s will cancel on the right and now we
must divide both sides by cos(45)
cos( 45) * x
15

cos( 45)
cos( 45)
15
x
cos( 45)
x  21.21
Solving for Sides VS Angles
• When solving for a
side you have to
use the function
cos, sin, and tan
• When solving for
an angle you need
to use the inverse
function
1
1
cos , sin , tan
1
(On your calculator
you must push 2nd
function first)
Solving for a Side
• Find the missing
side
12
x
30
y
• What are we going
to use to find x
SOH CAH TOA
x
sin( 30) 
12
x
sin( 30) *12  *12
12
sin( 30) *12  x
6x
Solving for a Side
• Find the missing
side
12
x
30
y
• What are we going
to use to find y
SOH CAH TOA
y
cos(30) 
12
y
cos(30) *12  *12
12
cos(30) *12  y
10.4  y