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TRIGONOMETRY
BASIC TRIANGLE STUDY:
RATIOS:
- SINE
- COSINE
- TANGENT
- ANGLES / SIDES
SINE LAW:
AREA OF A TRIANGLE:
-
GENERAL
- TRIGONOMETRY
- HERO’S
BASIC TRIANGLE STUDY
 Complimentary angles: 2 angles = 90°
 Supplementary angles: 2 angles = 180°
 Adjacent angles on the same line = 180°
 Opposite angles on the same line = each other
 The sum of the interior angles of a triangle = 180°
 Right triangles have one angle = 90°
 Pythagorean Theorem = a² + b² = c²
RATIOS
 “SOH” – “CAH” – “TOA”
 TRIGONOMETRIC RATIOS IN A RIGHT TRIANGLE
 SINE A =
𝑀𝐸𝐴𝑆𝑈𝑅𝐸 𝑂𝐹 𝑂𝑃𝑃𝑂𝑆𝐼𝑇𝐸 𝑆𝐼𝐷𝐸
𝑀𝐸𝐴𝑆𝑈𝑅𝐸 𝑂𝐹 𝐻𝑌𝑃𝑂𝑇𝐸𝑁𝑈𝑆𝐸
=
B
𝐴
𝐶
c
a
 COSINE A =
𝑀𝐸𝐴𝑆𝑈𝑅𝐸 𝑂𝐹 𝐴𝐷𝐽𝐴𝐶𝐸𝑁𝑇 𝑆𝐼𝐷𝐸
𝑀𝐸𝐴𝑆𝑈𝑅𝐸 𝑂𝐹 𝐻𝑌𝑃𝑂𝑇𝐸𝑁𝑈𝑆𝐸
=
𝐵
𝐶
C
A
 TANGENT A =
𝑀𝐸𝐴𝑆𝑈𝑅𝐸𝑀𝐸𝑁𝑇 𝑂𝐹 𝑂𝑃𝑃𝑂𝑆𝐼𝑇𝐸 𝑆𝐼𝐷𝐸
𝑀𝐸𝐴𝑆𝑈𝑅𝐸𝑀𝐸𝑁𝑇 𝑂𝐹 𝐴𝐷𝐽𝐴𝐶𝐸𝑁𝑇 𝑆𝐼𝐷𝐸
=
𝐴
𝐵
b
The adjacent side is the side next to the reference angle.
The opposite side is the side directly across from the reference angle.
Remember, it is important to understand that the names of the opposite
side and adjacent sides change when you move from one reference angle to the other.
RATIOS
m<A
SIN A
COS A
TAN A
0
1
0
20°
.342
.9397
.364
30°
.5
.866
.5774
45°
.7071
.7071
1
60°
.866
.5
1.7321
80°
.9848
.1736
5.6713
90°
1
0
0°
𝟏
𝟐
COS 30° =
√3
2
1
√2
COS 60° =
SIN 30° =
SIN 60° =
SIN 45° =
COS 45° =
√𝟑
𝟐
1
2
TAN 30° =
1
√2
TAN 45° = 1
𝟏
√𝟑
TAN 60° = √3
=
√𝟑
𝟑
SINE
B
𝑂𝑃𝑃𝑂𝑆𝐼𝑇𝐸
 SINE A =
𝐻𝑌𝑃𝑂𝑇𝐸𝑁𝑈𝑆𝐸
𝑎
 SINE A =
𝑐
c
a
C
b
A
COSINE
B
𝐴𝐷𝐽𝐴𝐶𝐸𝑁𝑇
 COS A =
𝐻𝑌𝑃𝑂𝑇𝐸𝑁𝑈𝑆𝐸
c
a
𝑏
 COS A =
𝑐
C
b
A
TANGENT
B
𝑂𝑃𝑃𝑂𝑆𝐼𝑇𝐸
 TAN A =
𝐴𝐷𝐽𝐴𝐶𝐸𝑁𝑇
c
𝑎
 TAN A =
𝑏
a
C
b
A
CALCULATOR
 CALCULATOR:
B
The button (key) SIN on the calculator
enables you to calculate the value of
SIN A if you know the measurement of
C
ANGLE A.
ie. SIN 30 = 0.5
The button (key) 𝐒𝐈𝐍 −𝟏 on the calculator enables you to
calculate the measure of the ANGLE A if you know SIN A
ie.
SIN −1
1
( ) = 30°
2
A
ANGLES / SIDES
FINDING MISSING SIDES USING TRIGONOMETRIC RATIOS
IN A RIGHT TRIANGLE,
 Finding the measure of x of side BC opposite to the known ANGLE A,
knowing the measure of the hypotenuse, requires the use of SIN A.
x
SIN 50° =
5
or
x = 5 · SIN 50° = 3.83
 Finding the measure of y of side AC adjacent to the known ANGLE A,
knowing the measure of the hypotenuse, requires the use of COS A.
COS 50° =
𝑦
5
or
A
y = 5 · COS 50° = 3.21
50⁰
5 cm
y
C
B
x
ANGLES / SIDES
 FINDING MISSING SIDES USING TRIGONOMETRIC RATIOS
IN A RIGHT TRIANGLE (CONTINUED),
 Finding the measure of x of side BC opposite to the known angle A,
knowing also the measure of the adjacent side to angle A, requires
the use of TAN A
A
x
TAN 30⁰ =
⇒ x = 4 · TAN 30⁰ = 2.31 cm
4
30⁰
4 cm
C
B
x
ANGLES / SIDES
 FINDING MISSING ANGLES USING TRIGONOMETRIC RATIOS
IN A RIGHT TRIANGLE,
 Finding the acute angle A when its opposite side and the hypotenuse
are known values require the use of SIN A.
4
SIN A = 4 ⇒ m ∠A = SIN¯¹ ( )= 53.1⁰
B
5
5
4
C
5
A
ANGLES / SIDES
 FINDING MISSING ANGLES USING TRIGONOMETRIC RATIOS
IN A RIGHT TRIANGLE,
 Finding the acute angle A when its adjacent side and the
hypotenuse are know values require the use of COS A
3
3
B
COS A =
⇒ m ∠ A = COS ¯¹ ( ) = 41.1⁰
4
4
4
C
A
3
ANGLES / SIDES
 FINDING MISSING ANGLES USING TRIGONOMETRIC RATIOS
IN A RIGHT TRIANGLE,
 Finding the acute angle A when its opposite side and adjacent side
are known values requires the use of TAN A
3
B
TAN A =
⇒ m ∠ A = TAN ¯¹ ( 3 )= 56.3⁰
2
2
3
C
A
2
SINE LAW
 The sides in a triangle are directly proportional to the SINE of
the opposite angles to these sides.
A
c
b
𝑎
𝑏
𝑐
=
=
B
𝑆𝐼𝑁 𝐴
𝑆𝐼𝑁 𝐵
𝑆𝐼𝑁 𝐶
C
a
 The SINE LAW can be used to find the measure of a missing
side or angle.
CASE 1: Finding a side when we know two angles and a side
We calculate the measure of x of AC
𝑥
15
15 SIN 50°
=
⟹𝑥=
= 13.27 cm A
𝑆𝐼𝑁 50°
𝑆𝐼𝑁 60°
𝑆𝐼𝑁 60°
60°
B
x
C
50°
15 cm
SINE LAW
The SINE LAW can be used to find the measure of a missing
side or angle.
CASE 2: Finding the angle when we know the two sides and
the opposite angle to one of these sides
We calculate the measure of angle B
10
13
10 𝑆𝐼𝑁 50°
=
⟹ 𝑆𝐼𝑁 𝐵 =
= 0.5893 ⟹ 𝑚 ∠ B = 36°
𝑆𝐼𝑁 𝐵
𝑆𝐼𝑁 50°
13
A
50°
B
x
10 cm
C
13 cm
AREA OF A TRIANGLE
GENERAL FORM
AREA =
𝐵𝐴𝑆𝐸 𝑥 𝐻𝐸𝐼𝐺𝐻𝑇
2
or
AREA =
𝐿𝐸𝑁𝐺𝑇𝐻 𝑥 𝑊𝐼𝐷𝑇𝐻
2
HEIGHT
L
W
BASE
AREA OF A TRIANGLE
TRIGONOMETRIC FORMULA
AREA =
AREA =
𝑎 𝑋 𝑐 𝑆𝐼𝑁 𝐵
2
A
𝑎 𝑋 𝑏 𝑆𝐼𝑁 𝐶
2
c
b
h
AREA =
𝑏 𝑋 𝑐 𝑆𝐼𝑁 𝐴
2
C
B
H
a
AREA OF A TRIANGLE
HERO’S FORMULA
 When you are given the measures for all three sides a,
b, c of a triangle, Hero’s Formula enables you to
calculate the area of a triangle.
AREA =
A
𝑝 (𝑝 − 𝑎)(𝑝 − 𝑏)(𝑝 − 𝑐)
b
c
 P = half the perimeter of the triangle
C
B
a
A
c
b
AREA OF A TRIANGLE
C
a
B
GENERAL
TRIGONOMETRIC
3.55 cm
12 𝑥 3.55
2
12 cm
= 21.3 𝑐𝑚2
A=
8 cm
6 cm
1
2
6 + 12 + 8 = 13
12 cm
A = 13(13 − 6)(13 − 12)(13 − 8)
A = 13 𝑥 7 𝑥 1 𝑥 5 =
26.4°
36.3°
HERO’S
P=
8 cm
6 cm
12 cm
A=
117.3°
455 = 21.3 𝑐𝑚2
12 𝑥 6 𝑥 𝑆𝐼𝑁 36.3°
2
= 21.3 𝑐𝑚2
A=
12 𝑥 8 𝑥 𝑆𝐼𝑁 26.4°
2
= 21.3𝑐𝑚2
A=
6 𝑥 8 𝑥 𝑆𝐼𝑁 117.3°
2
= 21.3 𝑐𝑚2