A PowerPoint presentation explaining the basics of trigonometry in a fun and exciting way1

1. TRIGONOTMETRY

2. Table of contents
1. 2.
3.
4. 5.
Basic trig SINE RULE
COSINE RULE (Side)
COSINE RULE (Angle) QUESTIONS

3. 1.
Basic trig

4. RIGHT ANGLED TRIANGLES
the side opposite
to the right angle
is called the
hypotenuse

5. RIGHT ANGLED TRIANGLES
hypotenuse
the side that is close
to or touching the
angle is called the
adjacent
the side that is
opposite the angle
is called the
opposite
θ

6. hypotenuse
adjacent
opposite
Sine is the ratio of the opposite to the hypotenuse
𝑠𝑖𝑛𝑒(𝜃) =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
Cosine is the ratio of the adjacent to the hypotenuse
𝑐𝑜𝑠𝑖𝑛𝑒(𝜃) =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
Tangent is the ratio of the opposite to the adjacent
𝑡𝑎𝑛𝑔𝑒𝑛𝑡(𝜃) =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
RIGHT ANGLED TRIANGLES
θ

7. H
A
O
A common mnemonic used to remember these
rules is
SOH CAH TOA
𝑺 =
𝑶
𝑯
𝑪 =
𝑨
𝑯
𝑻 =
𝑶
𝑨
RIGHT ANGLED TRIANGLES

8. 4 cm
hypotenuse
x
adjacent
6 cm
To find x:
The hypotenuse is given and the adjacent is
missing so, cosine is used
𝑐𝑜𝑠𝑖𝑛𝑒(𝜃) =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑐𝑜𝑠 32 =
𝑥
4
𝑐𝑜𝑠 32 4 = 𝑥
x = 3.34 cm
EXAMPLE
32̊
opposite
·

9. Sine inverse is used when both the opposite and the
hypotenuse are given BUT the angle is missing
Cosine inverse is used when both the adjacent and the
hypotenuse are given BUT the angle is missing
Tangent inverse is used when both the opposite and
the adjacent are given BUT the angle is missing
INVERSES
s𝑖𝑛𝑒−1
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
= 𝜃
cos𝑖𝑛𝑒−1
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
= 𝜃
2
𝑡𝑎𝑛𝑔𝑒𝑛𝑡−1
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
= 𝜃
SO THE INVERSES ARE USED TO FIND THE
ANGLE

10. 4 cm
hypotenuse
3.34 cm
adjacent
6 cm
To find x:
The hypotenuse is given and the adjacent is
given so, cosine inverse is used
cos𝑖𝑛𝑒−1 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
= 𝑎𝑛𝑔𝑙𝑒
cos−1
3.34
4
= 𝑥
x = 33.4º
EXAMPLE
xº
opposite

11. 2.
SINE RULE

12. Sine Rule
The sine rule enables you to calculate lengths
and angles in any triangle
There are two versions of this rule
(one for calculating the side and one for the angle)

13. Sine Rule
The sine rule is used to find:
a side when two angles and an opposite
side are known
an angle when two sides and an opposite
angle are known.

14. A
B C
b
a
c
Sine Rule for side
𝑎
𝑠𝑖𝑛 𝐴
=
𝑏
𝑠𝑖𝑛𝐵
=
𝑐
𝑠𝑖𝑛𝐶

15. Q
P R
9 cm
x cm
Example
75º 75º
To find x:
𝑝
𝑠𝑖𝑛𝑃
=
𝑟
𝑠𝑖𝑛𝑅
9
sin75
=
𝑥
𝑠𝑖𝑛75
9
sin75
× 𝑠𝑖𝑛75 = 𝑥
𝒙 = 𝟗 𝒄𝒎
We are given two sides and one angles and
so the sine rule can be applied

16. A
B C
b
a
c
Sine Rule for Angle
𝑠𝑖𝑛𝐴
𝑎
=
𝑠𝑖𝑛𝐵
𝑏
=
𝑠𝑖𝑛𝐶
𝑐

17. Q
P R
9 cm
4 cm
Example
75º xº
To find x:
𝑠𝑖𝑛𝑃
𝑝
=
𝑠𝑖𝑛𝑅
𝑟
sin75
9
=
𝑠𝑖𝑛𝑥
4
sin75 × 4
9
= 𝑠𝑖𝑛𝑥
0.429300 … = 𝑠𝑖𝑛𝑥
s𝑖𝑛−1
0.429300 = 𝑥
𝒙 = 𝟐𝟓. 𝟒 ̊̊
We are given two angles and one side and so
the sine rule can be used

18. 3.
COSINE RULE (SIDE)

19. COSINE RULE FOR Side
The cosine rule is a formula which can be used to
calculate the missing sides of a triangle or to find
a missing angle.
To do this we need to know the two
arrangements of the formula and what each
variable represents.

20. The cosine rule is used to find:
a side when two sides and the angle in between
them are known
an angle when three sides are known
COSINE RULE FOR Side

21. COSINE RULE FOR Side
A
B C
b
a
c
The cosine rule for sides is:
a²= b² + c² - (2bc + cosA)

22. Example
A
B C
7 cm
x cm
3 cm
To find x:
a²= b² + c² - (2bc + cosA)
a²=7²+3²- (2×3×7+ cos35)
a²= 23.5956 cm
a=
2
23.5956 cm
a= 4.86 cm
35º
The cosine rule is used because
there are 2 sides AND one angle

23. 4.
COSINE RULE (ANGLES)

24. COSINE RULE FOR Angles
A
B C
b
a
c
The cosine rule for angles is:
cos(A) =
b2 + c2 − a2
2bc

25. Example
A
B C
7 cm
5 cm
3 cm
To find x:
cos(A) =
b2 + c2 − a2
2bc
cos(A) =
72 + 32 − 52
2 × 7 × 3
cos−1(0.7857) = A
A = 38.2̊
xº
The cosine rule is used because
there are 3 sides AND an unknown
angle

26. 5.
QUESTIONS

36. THANK YOU!