Finding trigonometric ratio

1. MR. LUIS V.
SALENGA
(Teacher)

3. Drill
•Use paperless app

4. 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 ( right triangle).

5. EXPLAIN and USE the
relationship between the sine
and cosine of complementary
angles.

11. What are sin, cos and tan ?

12. What are sin cos and tan ?
They are the 3 ways that two sides can be
selected in relation to one angle.
Ɵ

13. What are sin , cos and tan ?
They are the 3 ways that two sides can be selected in
relation to one angle
Ɵ
O (opposite)

14. What are sin cos and tan ?
They are the 3 ways that two sides can be selected in
relation to one angle.
Ɵ
O (opposite)

15. What are sin, cos and tan ?
They are the 3 ways that two sides can be selected
in relation to one angle.
Ɵ
H (hypotenuse)

16. What are sin , cos and tan ?
They are the 3 ways that two sides can be
selected in relation to one angle
Ɵ
H (hypotenuse) O (opposite)

17. What are sin , cos and tan ?
Ɵ
A (adjacent)
O (opposite)
H (hypotenuse)

18. What are sin , cos and tan ?
Ɵ

19. What are sin, cos and tan ?
Ɵ H (hypotenuse)

20. What are sin, cos and tan ?
Ɵ H (hypotenuse)
O (opposite)

21. What are sin, cos and tan ?
Ɵ H (hypotenuse)
O (opposite)
A
(adjacent)

22. What are sin cos and tan ?
They are the 3 ways that an angle can
be found when given the lengths of two sides
-1 -1 -1
Ɵ

23. What are sin , cos and tan ?
They are the 3 ways that an angle can
be found when given the lengths of two sides
-1 -1 -1
Ɵ

24. Finding Trigo Ratios
A trigonometric ratio is a ratio of the
lengths of two sides of a right triangle.
The word trigonometry is derived from the
ancient Greek language and means
measurement of triangles. The three
basic trigonometric ratios are sine,
cosine, and tangent, which are
abbreviated as sin, cos, and tan
respectively.

25. Trig Ratios A
Hypotenuse
 Let ∆ABC be a right
triangle. The sine, the
cosine, and the tangent
of the acute angle A
are defined as follows.

26. Trig Ratios A
Hypotenuse
 Let ∆ABC be a right
triangle. The sine, the
cosine, and the tangent
of the acute angle A
are defined as follows.

27. Trig Ratios
 Let ∆ABC be a right
triangle. The sine, the
cosine, and the tangent
of the acute angle A
are defined as follows.
sin A =
side opposite A
hypotenuse
=
c
cos A =
side adjacent to A
hypotenuse
=
b
c
tan A =
side opposite A
side adjacent to A
=
a
b
a

28. Trig Ratios A
Hypotenuse

29. Trig Ratios A
Hypotenuse

30. The Trig Ratios
SohCahToa
sin A =
Side opposite A
hypotenuse
=
O
A
cos A =
Side adjacent to A
hypotenuse
=
tan A =
Side opposite A
Side adjacent to A
=
A
H
O
A
Hypotenuse
A

31. Ex. 1: Finding Trig Ratios
Find the sine, the
cosine, and the
tangent ratios for A
in each triangle
beside.
15
8
17
A
B
C

32. Ex. 1: Finding Trig Ratios
Find the sine, the
cosine, and the
tangent ratios for
A in each
triangle beside. 15
8
17
A
B
C
H
O

33. Ex. 1: Finding Trig Ratios
Find the sine,
the cosine, and
the tangent
ratios for A in
each triangle
beside.
15
8
17
A
B
C
h o
a

34. Ex. 1: Finding Trig Ratios
 Find the sine, the
cosine, and the
tangent ratios for A
in each triangle
beside.
15
8
17
A
B
C
Soh Cah Toa
Soh  Sine A =
o
h

35. Ex. 1: Finding Trig Ratios
 Find the sine, the
cosine, and the
tangent ratios for A
in each triangle
beside.
15
8
17
A
B
C
Soh Cah Toa
Soh  Sine A = =
o
h
8

36. Ex. 1: Finding Trig Ratios
 Find the sine, the
cosine, and the
tangent ratios for A
in each triangle
beside.
15
8
17
A
B
C
Soh Cah Toa
Soh  Sine A = =
o
h
8
17

37. Ex. 1: Finding Trig Ratios
 Find the sine, the
cosine, and the
tangent ratios for A
in each triangle
beside.
15
8
17
A
B
C
Soh Cah Toa
Cah  cos A =
a
h
=
15
17

38. Ex. 1: Finding Trig Ratios
 Find the sine, the
cosine, and the
tangent ratios for A
in each triangle
beside.
15
8
17
A
B
C
Soh Cah Toa
Toa  tan A =
o
a
=
8
15

39. Ex. 1: Finding Trig Ratios
15
8
17
A
B
C
sin A =
opposite
hypotenuse
cos A =
adjacent
hypotenuse
tan A =
opposite
adjacent
8
17
≈ 0.4706
15
17
≈
0.8824
8
15
≈
0.5333
Trig ratios are often
expressed as decimal
approximations.

40. 15
8
17
A
B
C
What are sin cos and tan ?
They are the 3 ways that an angle can
be found when given the lengths of two sides
-1 -1 -1
Ɵ

41. Let’s practice.
C
2cm
B 3cm A
Find the measure of the angle.
SOH CAH TOA
Ɵ
tan =
2
3
Ɵ
= 𝑡𝑎𝑛−1
(
2
3
)
Ɵ
= 33.69°
Ɵ
tan =
𝑜
𝑎
Ɵ

42. Let’s practice.
Find the measure of the angle.
SOH CAH TOA
33.69°
C
2cm
B 3cm A
sin 33.69 x h = 2
h=
2
sin 33.69
sin 33.69 =
2
ℎ
sin =
𝑜
ℎ
Ɵ
h= 3.61cm

43. Practice some more…
Find the measure of angle A
24.19 12
A 21
SOH CAH TOA
sin =
𝑂
ℎ
Ɵ
sin =
12
24.19
Ɵ
= cos−1
( 12/24.19)
Ɵ
= 29.74°
Ɵ

44. Practice some more.
SOH CAH TOA
cos =
𝑎
ℎ
Ɵ
cos 42 =
𝑎
14.5
cos 42 x 14.5 = a
a = 10.78 cm

45. Practice some more.
SOH CAH TOA
tan =
𝑜
𝑎
Ɵ
tan 38 =
𝑜
10
tan 38 x 10 = o
o = 7.81 cm

46. Indirect Measurement
 You are measuring the height of a
pine tree in Baguio City. You stand
15m from the base of the tree. You
measure the angle of elevation
from a point on the ground to the
top of the top of the tree to be 59°.
To estimate the height of the tree,
you can write a trigonometric ratio
that involves the height h and the
known length of 15m.
15m

47. Indirect Measurement
15m
SOH CAH TOA
tan =
𝑜
𝑎
Ɵ
tan =
𝑥
15
59
tan 59 x 15 = x
x = 24.96 cm
The tree is about 25m tall.

48. Estimating Distance
 Escalators. The escalator at
the Brisbane Central Rail
Station rises 21m at a
30° angle of elevation.
 To find the distance d a person
travels on the escalator stairs,
you can write a trig ratio that
involves the hypotenuse and
the known leg of 21m.
30°
d
21m

49. Estimating Distance
30°
d
21m
sin =
𝑂
ℎ
Ɵ
SOH CAH TOA
sin 30 =
21
𝑑
sin 30 x d = 21
d =
21
sin 30
d = 42 m
A person travels 42m on the escalator stairs.

50. Group Activity: Using the inverse Trig ratios to find
angles
The angle that an anchor line
makes with the seabed is really
critical for holding the ship. This
angle depends upon the type of
seabed, etc. In this case, the
seabed is mud and the best angle
for holding this ship is between
42-50 degrees.
Will the boat be safely anchored?

51. Group Activity: Using the inverse Trig ratios to find
angles
The angle that an anchor line makes with the
seabed is really critical for holding the ship.
This angle depends upon the type of seabed,
etc. In this case, the seabed is mud and the best
angle for holding this ship is between 42-50
degrees.
Will the boat be safely anchored?
SOH CAH TOA
sin =
𝑂
ℎ
Ɵ
sin =
𝑂
ℎ
Ɵ
sin =
18.88
30
Ɵ
=sin−1
(
18.88
30
)
Ɵ
= 39°
Ɵ
No, the boat will not be safely anchored

52. Ɵ
x = 8.39 cm X = 48.05 cm θ = 31.59°
θ= 37.69

53. Assignment:
When the space shuttle is 5 miles from
the runway, its glide angle is about 19°.
Find the shuttle’s altitude at this point in
its descent. Round your answer to the
nearest tenth.