Download Trigonometric Ratios - Effingham County Schools

Trigonometric Ratios
How do we use trig ratios?
M2 Unit 2: Day 3
In a Triangle, we know that the angles
have a sum of 180 and that the two
acute angles are complementary
q
So, if one angle is
q
Then the other one is
90 - q
90 - q
q
90 - q
Assume mÐX = q
X
then mÐY = 90 - q
what if: mÐX = 90 - q
then mÐX = q
Z
Y
Trigonometric Ratios:
Are ratios of the lengths of 2 sides of
a right triangle.
•There are 3 basic trig ratios: sine, cosine,
and tangent (abbreviated sin, cos, and tan)
•The value of a trig ratio depends only on
the measure of the acute angle, not on the
particular triangle being used to compute
the value.
SOHCAHTOA

If you can remember his name, then
you can remember your trig ratios!
Opposite of A
sin A 
Hypotenuse
Adjacent to A
cos A 
Hypotenuse
Opposite of A
tan A 
Adjacent to A



Opposite means “across from the
angle”
Adjacent means “attached to the
angle”
Hypotenuse is always opposite the
right angle.
Label the hypotenuse, opposite and
adjacent for angle A.
B
C
A
Label the hypotenuse, opposite and
adjacent for angle q .
q
Label the hypotenuse, opposite and
adjacent for angle X.
X
Z
Y
Now, label the hypotenuse, opposite
and adjacent for angle y.
opp 15
sin P 

 0.8824
hyp 17
adj 8
cos P 

 0.4706
hyp 17
opp 15
tan P 
  1.875
adj 8
Now find the sine, the cosine, and the tangent of Q .
opp 8
sin Q 

 0.4706
hyp 17
adj 15
cos Q 

 0.8824
hyp 17
opp 8
tan Q 
  0.5333
adj 15
•Notice
something about
the sine and
cosine ratios?
•How about the
tangent ratios?
2
13
5
opp 12
sin P 
  0.9231
hyp 13
adj 5
cos P 
  0.3846
hyp 13
opp 12
tan P 
  2.4
adj 5
12
Now find the sine, the cosine, and the tangent of Q .
opp 5
sin Q 
  0.3846
hyp 13
adj 12
cos Q 
  0.9231
hyp 13
opp 5
tan Q 
  0.4167
adj 12
•Notice
something about
the sine and
cosine ratios?
•How about the
tangent ratios?
Find tan q . Round to four decimal
places.
C
42
A
40
q
58
40
Tan q =
≈0.9524
42
B
Find sin 90 - q and tan q . Write each
answer as a decimal rounded to four
decimal places.
45
B
C
q
53
sin (90 - q) » .8491
tan (q) » .6222
28
A
You can use your calculator to find a
decimal approximation for trig ratios.
Example 3: Use your calculator to approximate
the given value to four decimal places.
a) sin 82°
Solutions:
0.9903
b) cos 30°
0.8660
c) tan 60°
1.7321
In summary, notice 4 things:
1.
2.
3.
4.
The 2 acute angles of a right
triangle are always complementary
The sin, cos, and tan of congruent
angles in similar triangles are
always equal no matter the side
lengths
The sin and cos ratios of 2
complementary angles are always
switched
The tan ratios of 2 complementary
angles are always reciprocals of
one another
sin q = cos (90 - q)
2
if sin q =
3
5
if sin A =
2
2
then cos(90 - q) =
3
5
then cos B =
2
cos q = sin (90 - q)
4
if cos q =
5
4
then sin(90 - q) =
5
6
if cos B =
7
6
then sin(90 - q) =
7
1
tan q =
tan (90 - q)
2
if tan q =
3
3
then tan(90 - q) =
2
8
if tan q =
5
5
then tan(90 - q) =
8
Homework:
 Page
159 (#1, 3, 7) and
 Page 166 (#2-14 even)