LAW OF SIGN AND COSINE

1. Trigonometry
Jinky S. Flores

2. Arrange Me
 Direction:
 The student will arrange the word.
 The answer should be send to the chat box
after the GO signal.
 The first one that can give the correct answer
will receive a reward.

3. WAL FO NESIS
 Clue: it is the relationship between the sides
and angles of non-right triangles.
 Answer: LAW OF SINES

4. WAL FO EINSOCS
 Clue: it relates the lengths of the sides of a
triangle to the cosine of one of its angles.
 Answer: LAW OF COSINES

5. QOUELIB NAGELRIT
 Clue: a triangle which does not contain a right
angle.
 Answer: OBLIQUE TRIANGLE

6. UCAET ALGITNER
Clue: a triangle that has an angle greater than
90°.
Answer: ACUTE TRIANGLE

7. SEBOUT LANRITGE
 Clue: a triangle that has an angle less than
90°.
 Answer: OBTUSE TRIANGLE

8. Introduction
In this section, we will solve
oblique triangles – triangles
that have no right angles.
As standard notation, the
angles of a triangle are labeled A, B, and C, and
their opposite sides are labeled a, b, and c.
To solve an oblique triangle, we need to know the
measure of at least one side and any two other
measures of the triangle—either two sides, two
angles, or one angle and one side.

9. Introduction
This breaks down into the following four cases:
1. Two angles and any side (AAS or ASA)
2. Two sides and an angle opposite one of them (SSA)
3. Three sides (SSS)
4. Two sides and their included angle (SAS)
The first two cases can be solved using the
Law of Sines, whereas the last two cases require the
Law of Cosines.

10. Introduction
 Law of sine – In any oblique triangles, the
sides are proportional to the sine of their
opposite angles or the sines of the angles is
proportional to their opposite sides.
B
c a
b
A C
𝑎
sin 𝐴
=
𝑏
sin 𝐵
=
𝑏
sin 𝐶
sin 𝐴
𝑎
=
sin 𝐵
𝑏
=
sin 𝐶
𝑐
or

11. Given Two Angles and One Side – AAS
For the triangle below C = 102, B = 29, and
b = 28 feet. Find the remaining angle and
sides.

12. Example AAS - Solution
The third angle of the triangle is
A = 180 – B – C
= 180 – 29 – 102
= 49.
By the Law of Sines, you have
.

13. Example AAS – Solution
Using b = 28 produces
and
cont’d

14. Example – SSA
 For the triangle below, a = 22 inches, b = 12
inches, and A = 42. Find the remaining side
and angles.

15. Example – Solution SSA
By the Law of Sines, you have
Reciprocal form
Multiply each side by b.
Substitute for A, a, and b.
Angle B is acute.
)
( 22
42
sin
12
sin 
B
)
(sin
sin
a
A
b
B 
a
A
b
B sin
sin

o
B 41
.
21


16. Example – Solution SSA
Now, you can determine that
C  180 – 42 – 21.41
= 116.59.
Then, the remaining side is
cont’d
)
59
.
116
sin(
)
42
sin(
22

c
C
A
a
c sin
sin

A
a
C
c
sin
sin

inches
40
.
29


17. Law of Cosines: Introduction
Two cases remain in the list of conditions
needed to solve an oblique triangle – SSS
and SAS.
If you are given three sides (SSS), or two sides
and their included angle (SAS), none of the
ratios in the Law of Sines would be complete.
In such cases, you can use the Law of
Cosines.

18. Law of Cosines
The law of cosines states that the square of the
length of one side is equal to the sum of the
squares of the other two sides minus the
product of twice of the two sides and the
cosines of the angle between them.
A
B
C
c a
b
C
abCos
b
a
c
B
acCos
c
a
b
A
bcCos
c
b
a
2
2
2
2
2
2
2
2
2
2
2
2










19. Example – Given two sides and
included angle (SAS)
Given: c = 4.5km, b= 6km, A = 150
 Find side a.
 𝑎2 = 𝑏2 + 𝑐2 − 2𝑏𝑐 𝑐𝑜𝑠𝐴
 𝑎2
= 62
+ 4.52
− 2 6 4.5 𝑐𝑜𝑠150°
 𝑎2
= 36 + 20.25 − 54 𝑐𝑜𝑠150°
 𝑎2 = 56.25 − 54 𝑐𝑜𝑠150°
 𝑎 = 103.02
 𝑎 = 10.15 km.

20. Example – Given two sides and
included angle (SAS)
Given: c = 4.5km, b= 6km, A = 150, 𝑎 = 10.15 km.
 To find the measure of angle B,
 𝑏2 = 𝑎2 + 𝑐2 − 2𝑎𝑏 𝑐𝑜𝑠𝐵
 62 = 10.152 + 4.52 − 2 10.15 4.5 𝑐𝑜𝑠𝐵
 36 = 103.0225 + 20.25 − 91.35 𝑐𝑜𝑠𝐵
 36 = 123.2725 − 91.35 𝑐𝑜𝑠𝐵
 91.35 𝑐𝑜𝑠𝐵 = 123.2725 − 36
 𝑐𝑜𝑠𝐵 =
87.2725
91.35
 𝐵 = 𝑐𝑜𝑠−1 87.2725
91.35
 𝐵 = 17.18°

21. Example: SAS
Given: A = 150°, B= 17.18°
To find the measure of angle C,
 C = 180° - A - B
 C = 180° - 150° - 17.18°
 C = 180° - 167.18°
 C = 12.82°
 The measure of angle C = 12.82°

22. Law of Cosines
Side, Side, Side
A
B
C
c a
b

23. Law of Cosines
 SSS
ab
c
b
a
C
ac
b
c
a
B
bc
a
c
b
A
2
cos
2
cos
2
cos
2
2
2
2
2
2
2
2
2










24. Example: Given the three sides (SSS)
 Given: a=55 cm, b= 70 cm, c= 35cm
Solve for angle A
cos 𝐴 =
𝑏2
+ 𝑐2
− 𝑎2
2𝑏𝑐
cos 𝐴 =
702 + 352 − 552
2(70)(35)
cos 𝐴 =
4900 + 1225 − 3025
4900
cos 𝐴 =
3100
4900
cos 𝐴 = 0.6327
𝐴 = 𝑐𝑜𝑠−1 (0.6327)
𝐴 = 50.75°

25. Example: SSS
 Given: a=55 cm, b= 70 cm, c= 35cm, 𝐴 = 50.75°
Solve for angle B
cos 𝐵 =
𝑎2
+ 𝑐2
− 𝑏2
2𝑎𝑐
cos 𝐵 =
552 + 352 − 702
2(55)(35)
cos 𝐵 =
3025 + 1225 − 4900
3850
cos 𝐵 =
−650
3850
cos 𝐵 = −0.1688
𝐵 = 𝑐𝑜𝑠−1 (-0.1688)
𝐵 = 99.72°

26. Example: SSS
Given: a=55 cm, b= 70 cm, c= 35cm, 𝐴 = 50.75°, 𝐵 = 99.72°
Solve for angle C
cos 𝐶 =
𝑎2 + 𝑏2 − 𝑐2
2𝑎𝑏
cos 𝐶 =
552 + 702 − 352
2(55)(70)
cos 𝐶 =
3025 + 4900 − 1225
7700
cos 𝐶 =
6700
7700
cos 𝐶 = 0.8701
𝐶 = 𝑐𝑜𝑠−1
(0.8701)
𝐶 = 29.53°

27. Thank You for Listening!!!