Transcript Trigonometric Ratios

Math 10
Ms. Albarico
Students are expected to:
•Demonstrate an understanding of and apply
properties to operations involving square roots.
• Relate the trigonometric functions to the ratios in
similar right triangles.
• Use calculators to find trigonometric values of
angles and angles to find when trigonometric
values are known.
* Solve problems using the trigonometric ratios.
TRIGONOMETRIC
RATIOS
A RATIO is a comparison
of two numbers. For
example;
boys to girls
cats : dogs
right : wrong.
In Trigonometry, the
comparison is between
sides of a triangle.
We need to do some
housekeeping before
we can proceed…
In trigonometry, the ratio we are talking about is the comparison of the sides of a
RIGHT TRIANGLE.
Two things MUST BE understood:
1. This is the hypotenuse.. This will
ALWAYS be the hypotenuse
2. This is 90°… this makes the right
triangle a right triangle…. Without it, we can
not do this trig… we WILL NOT use it in our
calculations because we COULD NOT do
calculations without it.
Now that we agree about the hypotenuse and
right angle, there are only 4 things left; the
2 other angles and the 2 other sides.
A
We will refer to the sides
in terms of their proximity
to the angle
hypotenuse
adjacent
opposite
If we look at angle A, there is
one side that is adjacent to it
and the other side is opposite
from it, and of course we have
the hypotenuse.
If we look at angle B, there is
one side that is adjacent to it
and the other side is opposite
from it, and of course we have
the hypotenuse.
hypotenuse
opposite
adjacent
B
Remember we won’t use
the right angle
X
One more thing…
θ this is the symbol for an unknown
angle measure.
It’s name is ‘Theta’.
Don’t let it scare you… it’s like ‘x’ except
for angle measure… it’s a way for us to
keep our variables understandable and
organized.
Here we
go!!!!
Trigonometric Ratios
Name
“say”
Abbreviation
Abbrev.
Ratio of an
angle measure
Sine
Cosine
tangent
Sin
Cos
Tan
Sinθ = opposite side
hypotenuse
cosθ = adjacent side
hypotenuse
tanθ =opposite side
adjacent side
Values of Trigonometric Function
00
300
450
600
900
Sine
0
0.5
1/2
3/2
1
Cosine
1
3/2
1/2
0.5
0
Tangent
0
1/ 3
1
3
Not defined
Cosecant
Not defined 2
2
2/ 3
1
Secant
1
2
2
Not defined
1
1/ 3
0
2/ 3
Cotangent Not defined 3
One more time…
Here are the ratios:
sinθ = opposite side
hypotenuse
cosθ = adjacent side
hypotenuse
tanθ = opposite side
adjacent side
Write the ratio for sin A
B
Sin A = a
c
c
Write the ratio for cos A
a
C
b
Cos A = b
c
A
Write the ratio for tan A
Let’s switch angles:
Find the sin, cos and
tan for Angle B:
Sin B = b
c
Tan A = a
b
Cos B = a
c
Tan B = b
a
Calculator Commands
Set your calculator to ‘Degree’…..
MODE (next to 2nd button)
Degree (third line down… highlight it)
2nd
Quit
Find tan A:
Tan A = opp/adj = 12/21
24.19
A
12
Tan A = .5714
21
Find tan A:
Tan A = 8/4 = 2
8
4
A
Note:
Given
Ratio of sides
Angle, side
Looking for
Use
Angle measure
SIN-1
COS-1
TAN-1
Missing side
SIN, COS, TAN
Calculator Commands Reminder
Set your calculator to ‘Degree’…..
MODE (next to 2nd button)
Degree (third line down… highlight it)
2nd
Quit
To solve for Angles:
tan x
o
opp

adj
C
hyp’
Now we need to look at
the two ratios involving
the hypotenuse:
sin xo =
Opposite
Hypotenuse
cos xo =
Adjacent
Hypotenuse
Opp’
xo
A
Adj’
B
Calculator Commands
 For Trigonometric Inverse Functions:
 1) Press 2ND, use
SIN for SIN-1
COS for COS-1
TAN for TAN-1
Calculate the angle b o below.
h
14.8cm
bo
9.7cm
(1) Identify the two sides marked.
a
adj
cos x 
hyp
(2) Choose the correct trig ratio .
9.7
o
cos b 
14.8
(3) Substitute in values .
o
cos b  0.655
o
b o  cos 1 0.655
b o = 49.1o
(4) Calculate the ratio(3 decimal
places).
(5) Use the inverse cosine function on
your calculator to calculate the angle .
Remembering the
Trigonometric Ratios:
Look again at the three trig ratios given below:
opp
sin x 
hyp
o
adj
cos x 
hyp
o
opp
tan x 
adj
o
Take the first letter of each word.
Write the letters in order.
S O H C A H T O A
Find an angle that has a
tangent (ratio) of 2
C
2cm
B
3
Round your answer to the
nearest degree.
3cm
A
Process:
I want to find an ANGLE.
I was given the sides (ratio).
Tangent is opp
adj
Solution:
TAN-1(2/3) = 34°
Ok… we’ve found side lengths, now
let’s find angle measures.
Refer to your table… what function will
we use to find angle measures?
SIN-1
COS-1
TAN-1
These are
called
INVERSE
FUNCTIONS.
Homework!
In your notebook, CYU # 18, 19, 20, 21, 22, 24,
and 25 on pages 239-240.
Class Work
In your notebook, solve the
following:
CYU # 12, 13, 14, 15, 16 on
pages 236-237.
Work Period
 Work with your group members about the
final design of your pet house.
 Remember :
Your scale drawing must be accurate and
precise.