Download trigonometric ratios

Defining Trigonometric Ratios
(5.8.1)
June 1st, 2016
*Since all right triangles already have one pair of
corresponding angles that are the same, it only
takes one more pair of congruent angles to make
the triangles similar by AA.
*Since similar triangles have proportional sides, this
means that all similar right triangles must have the
same side ratios.
bx b

ax a
cx c

bx b
cx c

ax a
*So given a particular side ratio in a triangle, we
should be able to determine the acute angle
measures of that triangle (given that all similar
triangles have the same angle measures).
6 2
 2
3 1
60
60
Trigonometric Ratios
The trigonometric functions (sine, cosine, tangent,
cosecant, secant, and cotangent) represent the
ratios of certain sides of right triangles from the
perspective of one of its acute angles. The sides
used in each ratio are identified by their position
from the angle of perspective, i.e., the side opposite
the angle, the side adjacent to the angle, and the
hypotenuse of the triangle.



SOH CAH TOA (for trig ratios)
Trigonometric Functions
-sine of angle

:
-cosecant of angle

opposite
sin  
hypotenuse

-cosine of angle
:
adjacent
cos 
hypotenuse

-tangent of angle
opposite
tan 
adjacent
Reciprocal
Trigonometric Functions
:
:
hypotenuse
csc 
opposite

-secant of angle
:
hypotenuse
sec 
adjacent

-cotangent of angle
adjacent
cot  
opposite
:
Ex: For the given triangle, set up and calculate each of the
following.
(a) The trigonometric ratios for the sine, cosine and tangent
A
of
.
(b) The trigonometric ratios for the cosecant, secant, and
A
cotangent of
.
(c) TheB
trigonometric ratios for the sine, cosine and tangent
of
.
(d) The trigonometric
ratios
for
the
cosecant,
secant,
and
B
cotangent of
.
opp 3
(a)sin A 
  0.6
hyp 5
adj 4
cos A 
  0.8
hyp 5
opp 3
tan A 
  0.75
adj 4
hyp 5
(b)csc A 
  1.667
opp 3
hyp 5
sec A 
  1.25
adj 4
adj 4
cot A 
  1.333
opp 3
hypotenuse
side opposite
A
side adjacent to A
opp 4
(c)sin B 
  0.8
hyp 5
adj 3
cos B 
  0.6
hyp 5
opp 4
tan B 
  1.333
adj 3
hyp 5
(d)csc B 
  1.25
opp 4
hyp 5
sec B 
  1.667
adj 3
adj 3
cot B 
  0.75
opp 4
hypotenuse
side adjacent
to B
side opposite B
Let’s say we knew thatmB  53.13
. Calculate
sin 53.13  0.8
cos 53.13  0.6
tan 53.13  1.333
*How do these answers compare to the
corresponding trigonometric
ratios
of
you
found
B
in part C?
They are the same!!!