for educ. purposes

2. Objectives:
• illustrate the trigonometric functions:
sine, cosine, tangent, secant, cosecant,
and cotangent.
• Solve the value of trigonometric ratios
from the given triangle and;
• Appreciate the importance of
trigonometry in real life situation

3. 3
Vocabulary Bank
comes from the Greek words trigonon, meaning
“three-sided,” and metron, meaning “measurement.”
It is the study of the relationships of the sides and the
angles of right triangles and the mathematical
properties of these relationships.
Trigonometry

4. Math Tale
The tallest tree is Hyperion, a coast
redwood (Sequoia sempervirens)
which is found in California. The tree is
380 ft. and 1 in. or 115.85 m tall
(Guinness World Record).
With this sky-towering height, have you
ever wondered how they managed to
do the measurement?

5. Trigonometric
Functions
sin 𝜃 =
𝑦
𝑟
𝑐𝑜𝑠𝜃 =
𝑥
𝑟
𝑡𝑎𝑛𝜃 =
𝑦
𝑥
csc 𝜃 =
𝑟
𝑦
𝑠𝑒𝑐𝜃 =
𝑟
𝑥
𝑡𝑎𝑛𝜃 =
𝑥
𝑦

6. Trigonometry
𝜃
Hypothenuse
Opposite
Adjacent

7. Trigonometric Ratios
The six trigonometric ratios are sine, cosine,
tangent, cosecant, secant, and cotangent. These
are defined as:
𝑠𝑖𝑛𝐴 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑙𝑒𝑔
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑐𝑠𝑐𝐴 =
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑐𝑜𝑠𝐴 =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑙𝑒𝑔
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑠𝑒𝑐𝐴 =
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑙𝑒𝑔
𝑡𝑎𝑛𝐴 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑙𝑒𝑔
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑙𝑒𝑔
𝑐𝑜𝑡𝐴 =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑙𝑒𝑔
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑙𝑒𝑔
SOH
CAH
TOA
CHO
SHA
CAO

8. Checkpoint 1:
Given the right triangle, identify the hypotenuse,
opposite leg, and adjacent leg to the labeled angle.
a. What is the opposite side of angle A?
b. What is the adjacent side of angle B?
c. What side is the hypothenuse?
a
a
c

9. Trigonometric Ratios Example 1
Consider the given triangle. Find for the trigonometric ratios for angle M.
𝑠𝑖𝑛𝑀 =
𝑜𝑝𝑝.
ℎ𝑦𝑝.
=
21
29
𝑐𝑠𝑐𝑀 =
ℎ𝑦𝑝.
𝑜𝑝𝑝.
=
29
21
𝑐𝑜𝑠𝑀 =
𝑎𝑑𝑗.
ℎ𝑦𝑝.
=
20
29
𝑠𝑒𝑐𝑀 =
ℎ𝑦𝑝.
𝑎𝑑𝑗.
=
29
20
𝑡𝑎𝑛𝑀 =
𝑜𝑝𝑝.
𝑎𝑑𝑗.
=
21
20
𝑐𝑜𝑡𝑀 =
𝑎𝑑𝑗.
𝑜𝑝𝑝.
=
20
21
M
20 21
29

10. Trigonometric Ratios Example 2
Consider the given triangle. Find for the trigonometric ratios for angle 𝐴.
62
+ 82
= 𝑐2
36 + 64 = 𝑐2
100 = 𝑐2
𝑐 = 10
𝑠𝑖𝑛𝐴 =
𝑜𝑝𝑝.
ℎ𝑦𝑝.
=
6
10
=
3
5
𝑐𝑠𝑐𝐴 =
ℎ𝑦𝑝.
𝑜𝑝𝑝.
=
10
6
=
5
3
𝑐𝑜𝑠𝐴 =
𝑎𝑑𝑗.
ℎ𝑦𝑝.
=
8
10
=
4
5
𝑠𝑒𝑐𝐴 =
ℎ𝑦𝑝.
𝑎𝑑𝑗.
=
10
8
=
5
4
𝑡𝑎𝑛𝐴 =
𝑜𝑝𝑝.
𝑎𝑑𝑗.
=
6
8
=
3
4
𝑐𝑜𝑡𝐴 =
𝑎𝑑𝑗.
𝑜𝑝𝑝.
=
8
6
=
4
3
In order to find all the ratios, the
hypotenuse needs to be known. By
Pythagoreantheorem,

11. Let us try this!
12
c

12. Let us try this!
In this example, sin 𝐵 =
5
13
. This means
that the adjacent side can be found
by Pythagorean theorem as
a. The opposite side is 5.
b. The adjacent side is 12
c. The hypotenuse is 13

14. Identify the hypotenuse, adjacent leg, and opposite leg in
reference to the given angle. Then, give the six
trigonometric ratios for the indicated angle.

15. Identify the hypotenuse, adjacent leg, and opposite leg in
reference to the given angle. Then, give the six
trigonometric ratios for the indicated angle.
Hypotenuse = 17
Adjacent leg =15
Opposite leg= 8
𝑠𝑖𝑛𝐵 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑙𝑒𝑔
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
=
8
17
𝑐𝑠𝑐𝐵 =
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
=
17
8
𝑐𝑜𝑠𝐵 =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑙𝑒𝑔
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
=
15
17
𝑠𝑒𝑐𝐵 =
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑙𝑒𝑔
=
17
15
𝑡𝑎𝑛𝐵 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑙𝑒𝑔
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑙𝑒𝑔
=
8
15
𝑐𝑜𝑡𝐵 =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑙𝑒𝑔
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑙𝑒𝑔
=
15
8

16. 16
A. Give the values of the six trigonometric ratios for each
of the following indicated angles.
1. 2.

17. Assignment
✘ Read and study the trigonometric ratio of
special angles
17

18. MATH TALE
The tallest tree is Hyperion, a coast redwood
(Sequoia sempervirens) which is found in
California. The tree is 380 ft. and 1 in. or 115.85 m
tall (Guinness World Record).
With this sky-towering height, have you ever
wondered how they managed to do the
measurement?
Laying a measuring tape would not surely be a good idea. This unit will provide us a
technique to do various measurements that would not be possible using ordinary
tools. We can apply the trigonometric ratios in this example; to measure the tallest tree
or even the tallest mountain.

19. 19

22. THANK You S0 MUCH !