4 ESO Academics - UNIT 07 - TRIGONOMETRY.

1. Unit 07 February
1. DEFINITION OF TRIGONOMETRY.
Trigonometry (from Greek trigonom “triangle” + metron “measure”) is a
branch of mathematics that studies Triangles and the relationships between their
sides and the angles between these sides.
Trigonometry defines the Trigonometric Functions, which describe those
relationships and have applicability to cyclical phenomena, such as Waves. It is also the
foundation of the practical art of Surveying.
Axel Cotón Gutiérrez Mathematics 4º ESO 4.7.1

2. Unit 07 February
MATH VOCABULARY: Trigonometry, Triangle, Trigonometric Functions, Wave,
Surveying, Angle.
2. UNITS OF MEASUREMENT FOR ANGLES.
There are two commonly used units of measurement for angles. The more
familiar unit of measurement is that of Degrees. A circle is divided into 𝟑𝟑𝟑𝟑𝟑𝟑 equal
degrees, so that a right angle is 𝟗𝟗𝟗𝟗°.
The other common measurement for angles is Radians. One Radian is the angle
subtended at the center of a circle by an arc that is equal in length to the radius of the
circle.
Axel Cotón Gutiérrez Mathematics 4º ESO 4.7.2

3. Unit 07 February
Since the length of a the circumference is 𝟐𝟐𝟐𝟐𝟐𝟐:
𝟑𝟑𝟑𝟑𝟑𝟑° = 𝟐𝟐𝟐𝟐 𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓 𝒓𝒓 𝒓𝒓𝒓𝒓
Therefore:
MATH VOCABULARY: Degree, Right Angle, Radian.
3. TRIGONOMETRIC RATIOS OF AN ANGLE.
When we draw a perpendicular line to one side of the angle 𝜶𝜶 we get a right-
angled triangle. The legs of this triangle are called Opposite and Adjacent.
Axel Cotón Gutiérrez Mathematics 4º ESO 4.7.3

4. Unit 07 February
Opposite Leg is opposite the angle 𝜶𝜶, and Adjacent Leg is adjacent (next) to
angle 𝜶𝜶. There are six ways to form ratios of the three sides of this triangle, and each
of these ratios has a name:
Axel Cotón Gutiérrez Mathematics 4º ESO 4.7.4

5. Unit 07 February
If we drew other different right triangles for the same angle, what would
happen to the trigonometric ratios?
in 𝑨𝑨𝑨𝑨𝑨𝑨 in 𝑨𝑨𝑨𝑨’𝑪𝑪’ in 𝑨𝑨𝑨𝑨’’𝑪𝑪’’
𝒔𝒔𝒊𝒊 𝒏𝒏 𝜶𝜶
𝑩𝑩𝑩𝑩����
𝑨𝑨𝑨𝑨����
𝑩𝑩′𝑪𝑪′������
𝑨𝑨𝑨𝑨′�����
𝑩𝑩′′𝑪𝑪′′�������
𝑨𝑨𝑨𝑨′�����
Since the triangles 𝑨𝑨𝑨𝑨𝑨𝑨, 𝑨𝑨𝑨𝑨’𝑪𝑪’ and 𝑨𝑨𝑨𝑨’’𝑪𝑪’’ are similar:
𝑩𝑩𝑩𝑩����
𝑨𝑨𝑨𝑨����
=
𝑩𝑩′𝑪𝑪′�����
𝑨𝑨𝑨𝑨′�����
=
𝑩𝑩′′𝑪𝑪′′�������
𝑨𝑨𝑨𝑨′�����
Therefore, the value of 𝒔𝒔𝒔𝒔𝒔𝒔 𝜶𝜶 does not depend on the right triangle that we
use. The same thing can be said about the other trigonometric ratios.
MATH VOCABULARY: Trigonometric ratios, Hypotenuse, opposite Leg, Adjacent Leg,
Sine, Cosine, Tangent, Secant, Cosecant, Cotangent.
Axel Cotón Gutiérrez Mathematics 4º ESO 4.7.5

6. Unit 07 February
4. BASIC RELATIONS BETWEEN THE TRIGONOMETRIC RATIOS OF AN
ANGLE.
The trigonometric ratios of angle are not independent. They have some
relations between them. The Fundamental Relationships between the trigonometric
ratios of an angle are:
𝒕𝒕𝒕𝒕𝒕𝒕 𝜶𝜶 =
𝒔𝒔𝒔𝒔𝒔𝒔 𝜶𝜶
𝒄𝒄𝒄𝒄𝒄𝒄 𝜶𝜶
PROOF:
tan α =
sin α
cos α
=
𝑎𝑎
𝑏𝑏
𝑐𝑐
𝑏𝑏
=
𝑎𝑎 ∙ 𝑏𝑏
𝑐𝑐 ∙ 𝑏𝑏
=
𝑎𝑎
𝑐𝑐
= tan 𝛼𝛼
𝒔𝒔𝒔𝒔𝒔𝒔𝟐𝟐
𝜶𝜶 + 𝒄𝒄𝒄𝒄𝒄𝒄𝟐𝟐
𝜶𝜶 = 𝟏𝟏
PROOF:
sen2
α + cos2
α = �
𝑎𝑎
𝑏𝑏
�
2
+ �
𝑐𝑐
𝑏𝑏
�
2
=
𝑎𝑎2
𝑏𝑏2
+
𝑐𝑐2
𝑏𝑏2
=
𝑎𝑎2
+ 𝑐𝑐2
𝑏𝑏2
=
𝑏𝑏2
𝑏𝑏2
= 1
𝒕𝒕𝒕𝒕𝒕𝒕𝟐𝟐
𝜶𝜶 + 𝟏𝟏 = 𝒔𝒔𝒔𝒔𝒔𝒔𝟐𝟐
𝜶𝜶
PROOF: If we divide the second equation by 𝑐𝑐𝑐𝑐𝑐𝑐2
𝛼𝛼:
sen2
α + cos2
α
cos2α
=
𝑠𝑠𝑠𝑠𝑠𝑠2
𝛼𝛼
cos2α
+
cos2
α
cos2α
=
1
cos2α
= 𝑡𝑡𝑡𝑡𝑡𝑡2
𝛼𝛼 + 1 = 𝑠𝑠𝑠𝑠𝑠𝑠2
𝛼𝛼
Axel Cotón Gutiérrez Mathematics 4º ESO 4.7.6

7. Unit 07 February
5. TRIGONOMETRIC RATIOS OF 30°, 45° AND 60°.
If we have 𝟒𝟒𝟒𝟒°, the two legs have the same length 𝒍𝒍:
We have:
𝒂𝒂 = �𝒍𝒍𝟐𝟐 + 𝒍𝒍𝟐𝟐 = √𝟐𝟐 ∙ 𝒍𝒍
𝒔𝒔𝒔𝒔𝒔𝒔 𝟒𝟒𝟒𝟒° =
𝒍𝒍
𝒂𝒂
=
𝒍𝒍
√𝟐𝟐 ∙ 𝒍𝒍
=
𝟏𝟏
√𝟐𝟐
=
√𝟐𝟐
𝟐𝟐
= 𝒄𝒄𝒄𝒄𝒄𝒄 𝟒𝟒𝟒𝟒°
𝒕𝒕𝒕𝒕𝒕𝒕 𝟒𝟒𝟒𝟒° =
𝒔𝒔𝒔𝒔𝒔𝒔 𝟒𝟒𝟒𝟒°
𝒄𝒄𝒄𝒄𝒄𝒄 𝟒𝟒𝟒𝟒°
= 𝟏𝟏
For calculate the trigonometric ratios of 𝟑𝟑𝟑𝟑° and 𝟔𝟔𝟔𝟔°, we take the half of an
equilateral triangle of side 𝒍𝒍:
Firstly we calculate 𝒉𝒉:
Axel Cotón Gutiérrez Mathematics 4º ESO 4.7.7

8. Unit 07 February
𝒉𝒉 = �𝒍𝒍𝟐𝟐 − �
𝒍𝒍
𝟐𝟐
�
𝟐𝟐
= �
𝟑𝟑𝒍𝒍𝟐𝟐
𝟒𝟒
=
√𝟑𝟑
𝟐𝟐
∙ 𝒍𝒍
Therefore:
𝒔𝒔𝒔𝒔𝒔𝒔 𝟑𝟑𝟑𝟑° =
𝒍𝒍
𝟐𝟐
𝒍𝒍
=
𝟏𝟏
𝟐𝟐
= 𝒄𝒄𝒄𝒄𝒄𝒄 𝟔𝟔𝟔𝟔°
𝒄𝒄𝒄𝒄𝒄𝒄 𝟑𝟑𝟑𝟑° =
√𝟑𝟑
𝟐𝟐
𝒍𝒍
𝒍𝒍
=
√𝟑𝟑
𝟐𝟐
= 𝒔𝒔𝒔𝒔𝒔𝒔 𝟔𝟔𝟔𝟔°
𝒕𝒕𝒕𝒕𝒕𝒕 𝟑𝟑𝟑𝟑° =
𝒔𝒔𝒔𝒔𝒔𝒔 𝟑𝟑𝟑𝟑°
𝒄𝒄𝒄𝒄𝒄𝒄 𝟑𝟑𝟑𝟑°
=
𝟏𝟏
√𝟑𝟑
=
√𝟑𝟑
𝟑𝟑
𝒕𝒕𝒕𝒕𝒕𝒕 𝟔𝟔𝟔𝟔° =
𝒔𝒔𝒔𝒔𝒔𝒔 𝟔𝟔𝟔𝟔°
𝒄𝒄𝒄𝒄𝒄𝒄 𝟔𝟔𝟔𝟔°
= √𝟑𝟑
6. TRIGONOMETRIC RATIOS: FROM 0° TO 360°.
We can represent angles in a circumference which centre is the Origin of
Coordinates:
• Take the origin of coordinates as a Vertex of the triangle.
• Use the radius on the positive Semiaxis of abscissas as origin of the angle.
• Draw the extreme of the angle by measuring it:
If the radius is one, the circumference is called Goniometric Circumference.
Axel Cotón Gutiérrez Mathematics 4º ESO 4.7.8

9. Unit 07 February
If we represent an angle 𝜶𝜶 in this circumference, we get the point 𝑷𝑷(𝒙𝒙, 𝒚𝒚). This
point is the intersection of the side of the angle that is not on the x-axis with the
circumference.
𝑷𝑷(𝒙𝒙, 𝒚𝒚) is 1 unit away from the origin. Therefore, the sine of 𝜶𝜶 is the value of
the ordinate of P, and the cosine of 𝜶𝜶 is the value of the abscissa of P:
𝒄𝒄𝒄𝒄𝒄𝒄 𝜶𝜶 = 𝒙𝒙
𝒔𝒔𝒔𝒔𝒔𝒔 𝜶𝜶 = 𝒚𝒚
𝒕𝒕𝒕𝒕𝒕𝒕 𝜶𝜶 =
𝒚𝒚
𝒙𝒙
MATH VOCABULARY: Origin of Coordinates, Vertex, Semiaxis, Abscissas,
Counterclockwise, Clockwise, Goniometric Circumference.
Axel Cotón Gutiérrez Mathematics 4º ESO 4.7.9

10. Unit 07 February
7. SIGNS TRIGONOMETRIC RATIOS. QUADRANTS.
The Coordinate Axes divide the plane into four equal parts called Quadrants.
𝑰𝑰𝑰𝑰 𝟎𝟎° < 𝜶𝜶 < 𝟗𝟗𝟗𝟗° ⇒ 𝜶𝜶 ∈ 𝟏𝟏𝒔𝒔𝒕𝒕 𝒒𝒒𝒖𝒖𝒂𝒂𝒅𝒅𝒓𝒓𝒂𝒂𝒏𝒏𝒏𝒏
𝑰𝑰𝑰𝑰 𝟗𝟗𝟗𝟗° < 𝜶𝜶 < 𝟏𝟏𝟏𝟏𝟏𝟏° ⇒ 𝜶𝜶 ∈ 𝟐𝟐𝒏𝒏𝒅𝒅 𝒒𝒒𝒒𝒒𝒒𝒒𝒒𝒒𝒒𝒒𝒒𝒒𝒒𝒒𝒒𝒒
𝑰𝑰𝑰𝑰 𝟏𝟏𝟖𝟖𝟖𝟖° < 𝜶𝜶 < 𝟐𝟐𝟐𝟐𝟐𝟐° ⇒ 𝜶𝜶 ∈ 𝟑𝟑𝒓𝒓𝒅𝒅 𝒒𝒒𝒖𝒖𝒂𝒂𝒅𝒅𝒓𝒓𝒂𝒂𝒏𝒏𝒏𝒏
𝑰𝑰𝑰𝑰 𝟐𝟐𝟐𝟐𝟐𝟐° < 𝜶𝜶 < 𝟑𝟑𝟑𝟑𝟑𝟑° ⇒ 𝜶𝜶 ∈ 𝟒𝟒𝒕𝒕𝒉𝒉 𝒒𝒒𝒖𝒖𝒂𝒂𝒅𝒅𝒓𝒓𝒂𝒂𝒏𝒏𝒏𝒏
7.1. FIRST QUADRANT.
𝒔𝒔𝒔𝒔𝒔𝒔 𝜶𝜶 > 𝟎𝟎 𝒂𝒂𝒂𝒂𝒂𝒂 𝐜𝐜𝐜𝐜𝐜𝐜 𝜶𝜶 > 𝟎𝟎 ⇒ 𝐭𝐭𝐭𝐭 𝐭𝐭 𝜶𝜶 > 𝟎𝟎
Axel Cotón Gutiérrez Mathematics 4º ESO 4.7.10

11. Unit 07 February
7.2. SECOND QUADRANT.
𝒔𝒔𝒔𝒔𝒔𝒔 𝜶𝜶 > 𝟎𝟎 𝒂𝒂𝒏𝒏𝒅𝒅 𝐜𝐜𝐜𝐜𝐜𝐜 𝜶𝜶 < 𝟎𝟎 ⇒ 𝐭𝐭𝐭𝐭 𝐭𝐭 𝜶𝜶 < 𝟎𝟎
7.3. THIRD QUADRANT.
𝒔𝒔𝒔𝒔𝒔𝒔 𝜶𝜶 < 𝟎𝟎 𝒂𝒂𝒂𝒂𝒂𝒂 𝐜𝐜𝐜𝐜𝐜𝐜 𝜶𝜶 < 𝟎𝟎 ⇒ 𝐭𝐭𝐭𝐭 𝐭𝐭 𝜶𝜶 > 𝟎𝟎
Axel Cotón Gutiérrez Mathematics 4º ESO 4.7.11

12. Unit 07 February
7.4. FOURTH QUADRANT.
𝒔𝒔𝒔𝒔𝒔𝒔 𝜶𝜶 < 𝟎𝟎 𝒂𝒂𝒂𝒂𝒂𝒂 𝐜𝐜𝐜𝐜𝐜𝐜 𝜶𝜶 > 𝟎𝟎 ⇒ 𝐭𝐭𝐭𝐭 𝐭𝐭 𝜶𝜶 < 𝟎𝟎
Axel Cotón Gutiérrez Mathematics 4º ESO 4.7.12

13. Unit 07 February
7.5. REDUCE TO THE FIRST QUADRANT.
Let 𝜶𝜶 be an angle in the first quadrant. Then, 𝟏𝟏𝟏𝟏𝟏𝟏° − 𝜶𝜶 is an angle in the
second quadrant, 𝟏𝟏𝟏𝟏𝟏𝟏° + 𝜶𝜶 is an angle in the third quadrant, and 𝟑𝟑𝟑𝟑𝟑𝟑° − 𝜶𝜶 is an
angle in the fourth quadrant. The trigonometric ratios of 𝟏𝟏𝟏𝟏𝟏𝟏° − 𝜶𝜶 , 𝟏𝟏𝟏𝟏𝟏𝟏° + 𝜶𝜶 and
𝟑𝟑𝟑𝟑𝟑𝟑° − 𝜶𝜶 can be expressed in terms of the trigonometric ratios of 𝜶𝜶 .
𝒔𝒔𝒔𝒔𝒔𝒔 (𝟏𝟏𝟏𝟏𝟏𝟏° − 𝜶𝜶) = 𝒔𝒔𝒔𝒔𝒔𝒔 𝜶𝜶
𝒄𝒄𝒄𝒄𝒄𝒄 (𝟏𝟏𝟏𝟏𝟏𝟏° − 𝜶𝜶) = −𝒄𝒄𝒄𝒄𝒄𝒄 𝜶𝜶
𝒕𝒕𝒕𝒕𝒕𝒕 (𝟏𝟏𝟏𝟏𝟏𝟏° − 𝜶𝜶) = −𝒕𝒕𝒕𝒕𝒕𝒕 𝜶𝜶
𝒔𝒔𝒔𝒔𝒔𝒔 (𝟏𝟏𝟏𝟏𝟏𝟏° + 𝜶𝜶) = −𝒔𝒔𝒔𝒔𝒔𝒔 𝜶𝜶
𝒄𝒄𝒄𝒄𝒄𝒄 (𝟏𝟏𝟏𝟏𝟏𝟏° + 𝜶𝜶) = −𝒄𝒄𝒄𝒄𝒄𝒄 𝜶𝜶
𝒕𝒕𝒕𝒕𝒕𝒕 (𝟏𝟏𝟏𝟏𝟏𝟏° + 𝜶𝜶) = 𝒕𝒕𝒕𝒕𝒕𝒕 𝜶𝜶
Axel Cotón Gutiérrez Mathematics 4º ESO 4.7.13

14. Unit 07 February
𝒔𝒔𝒔𝒔𝒔𝒔 (𝟑𝟑𝟑𝟑𝟎𝟎° − 𝜶𝜶) = −𝒔𝒔𝒔𝒔𝒔𝒔 𝜶𝜶
𝒄𝒄𝒄𝒄𝒄𝒄 (𝟑𝟑𝟑𝟑𝟎𝟎° − 𝜶𝜶) = 𝒄𝒄𝒄𝒄𝒄𝒄 𝜶𝜶
𝒕𝒕𝒕𝒕𝒕𝒕 (𝟑𝟑𝟑𝟑𝟎𝟎° − 𝜶𝜶) = −𝒕𝒕𝒕𝒕𝒕𝒕 𝜶𝜶
MATH VOCABULARY: Coordinate Axes, Quadrants.
8. TRIGONOMETRIC RATIOS OF ANY ANGLE.
8.1. TRIGONOMETRIC RATIOS OF COMPLEMENTARY ANGLES.
Two angles are Complementary when they add up to 90 degrees.
Axel Cotón Gutiérrez Mathematics 4º ESO 4.7.14

15. Unit 07 February
𝒔𝒔𝒔𝒔𝒔𝒔 (𝟗𝟗𝟎𝟎° − 𝒙𝒙) = 𝒄𝒄𝒐𝒐𝒐𝒐 𝒙𝒙
𝒄𝒄𝒄𝒄𝒄𝒄 (𝟗𝟗𝟎𝟎° − 𝒙𝒙) = 𝒔𝒔𝒔𝒔𝒔𝒔 𝒙𝒙
𝒕𝒕𝒕𝒕𝒕𝒕 (𝟗𝟗𝟎𝟎° − 𝒙𝒙) =
𝟏𝟏
𝒕𝒕𝒕𝒕𝒕𝒕 𝒙𝒙
8.2. TRIGONOMETRIC RATIOS OF SUPPLEMENTARY ANGLES.
Two Angles are Supplementary when they add up to 180 degrees.
𝑰𝑰𝑰𝑰 𝜶𝜶 + 𝜷𝜷 = 𝟏𝟏𝟏𝟏𝟏𝟏° ⇒ 𝜷𝜷 = 𝟏𝟏𝟏𝟏𝟏𝟏 − 𝜶𝜶
We will have the same result as reducing from second to first quadrant:
Axel Cotón Gutiérrez Mathematics 4º ESO 4.7.15

16. Unit 07 February
8.3. TRIGONOMETRIC RATIOS OF CONJUGATE ANGLES.
Two angles that sum to a complete angle (1 turn, 360°, or 2π radians) are
called Explementary Angles or Conjugate Angles.
𝑰𝑰𝑰𝑰 𝜶𝜶 + 𝜷𝜷 = 𝟑𝟑𝟑𝟑𝟎𝟎° ⇒ 𝜷𝜷 = 𝟑𝟑𝟔𝟔𝟎𝟎 − 𝜶𝜶
We will have the same result as reducing from fourth to first quadrant:
Axel Cotón Gutiérrez Mathematics 4º ESO 4.7.16

17. Unit 07 February
8.4. TRIGONOMETRIC RATIOS OF NEGATIVE ANGLES.
It is almost the same case as above. Negative Angles are a way of measuring an
angle from a different direction.
𝒔𝒔𝒔𝒔𝒔𝒔 (−𝜶𝜶) = − 𝒔𝒔𝒔𝒔𝒔𝒔 𝜶𝜶
𝒄𝒄𝒐𝒐𝒐𝒐 (−𝜶𝜶) = 𝒄𝒄𝒄𝒄𝒄𝒄 𝜶𝜶
𝒕𝒕𝒕𝒕𝒕𝒕 (−𝜶𝜶) = − 𝒕𝒕𝒕𝒕𝒕𝒕 𝜶𝜶
8.5. TRIGONOMETRIC RATIOS OF ANGLES GREATER THAN 360°.
In general, the trigonometric ratios of 𝒏𝒏 ⋅ 𝟑𝟑𝟑𝟑𝟑𝟑° + , where𝜶𝜶 𝒏𝒏 ∈ ℤ and
𝟎𝟎° ≤ 𝜶𝜶 < 𝟑𝟑𝟑𝟑𝟑𝟑° are the same as the trigonometric ratios of 𝜶𝜶 We divide the angle by.
360 and the remainder will be 𝜶𝜶.
Axel Cotón Gutiérrez Mathematics 4º ESO 4.7.17

18. Unit 07 February
MATH VOCABULARY: Complementary Angles, Supplementary Angles, Conjugate
Angles, Explementary Angles.
9. SOLVING TRIANGLES
Solving a Triangle means to find the lengths of all its sides and the measures of
all its angles. The measures of the angles and the lengths of the sides are related to
one another. If you know the measure (length) of three out of the six parts of the
triangle (at least one side must be included), you can find the measures of the
remaining sides an angles.
If the triangle is a right triangle, you can use simple trigonometric ratios to find
the missing parts. In a general triangle (acute or obtuse), you need to use other
techniques, including the Law of Cosines and the Law of Sines. You can also find the
area of triangles by using trigonometric ratios.
The triangle can be located on a plane or on a sphere. This problem often
occurs in various trigonometric applications, such as geodesy, astronomy,
construction, navigation, etc.
Axel Cotón Gutiérrez Mathematics 4º ESO 4.7.18