NOTES IN TRIGONOMETRY
Triangles are classified in terms of their
Angles and its Measure:
Angle is the space between intersecting
rays or lines. The point of intersection is called
interior angles and the sides.
➢ For angles as reference:
o Right triangle - triangle with one
interior angle equal to 900.
vertex. Angles are positive when measured
counterclockwise
(ccw)
and
negative
in
o
Oblique triangle - triangles are
classified into:
clockwise (cw) direction.
▪
vertex
Acute Triangle - triangles
with one interior angle
Angle
equal to 90°.
▪
Obtuse
Triangle
-
triangles in which one of
Units Used in Measuring Angles:
the interior angle is more
1 revolution = 360 degrees
than 90°but less than
= 2Π radian
1800.
= 400 grads
▪
= 400 gons
Equiangular
Triangle
-
triangles in which all of
= 6400 mils
the
interior
angle
are
equal.
➢ For sides as reference:
o Isosceles Triangle - triangle with
zero angle
Angle Equivalent
in Degrees
θ= 00
acute angle
00 < θ< 900
right angle
θ= 900
obtuse angle
00 < θ< 1800
straight angle
θ= 1800
Reflex angle
1800 < θ< 3600
with all sides are equal. Also
full angle or Perigon
θ= 3600
called equiangular Triangle.
Names
Sum of Angles:
A + B = 90˚
A and B are complementary
θ + α= 180˚ θ and α are supplementary
β + γ= 360˚ β and γ are explementary
Note: The following expressions are usually
two sides equal.
o
Scalene Triangle - triangle with
none of the sides are equal.
o
Equilateral Triangle
-
triangle
Schwartz’s Inequality:
The sum of any two sides of any triangle
is greater than the third side.
𝒂 + 𝒃 > 𝒄: 𝒃 + 𝒄 > 𝒂; 𝒂 + 𝒄 > 𝒃
c
B
a
used to some problems:
Complement of A = 90°- A
Supplement of B = 180°- B
A
C
b
Explement of C = 360°- C
Angle of Depression and Elevation:
Classifications of Triangles:
prepared by: engr.apn
The angle of depression is the angle
from horizontal down to the line of sight from
csc 𝜃 =
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝒄
=
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 𝒂
elevation is the angle from the horizontal up
TRIGONOMETRIC IDENTITIES
Co-function Relations:
sin(90 − 𝜃) = cos 𝜃
to the line of sight from the observer to an
cos(90 − 𝜃) = sin 𝜃
object above. The angle of elevation is equal to
tan(90 − 𝜃) = cot 𝜃
the angle of depression.
cot(90 − 𝜃) = tan 𝜃
the observer to an object below. The angle of
sec(90 − 𝜃) = csc 𝜃
csc(90 − 𝜃) = sec 𝜃
RIGHT TRIANGLES
Solution of Right Triangles:
Pythagorean Theorem - the square of the
hypotenuse is equal to the sum of the
squares of the other two sides.
Reciprocal Relations:
1
1
sin 𝜃 =
cot 𝜃 =
csc 𝜃
tan 𝜃
1
1
cos 𝜃 =
sec 𝜃 =
sec 𝜃
cos 𝜃
1
1
tan 𝜃 =
csc 𝜃 =
cot 𝜃
sin 𝜃
Tangent and Co-tangent Relations:
sin 𝜃
cos 𝜃
tan 𝜃 =
cot 𝜃 =
cos 𝜃
s𝜃
Pythagorean Relations:
sin2 𝜃 + cos 2 𝜃 = 1
1 + cot 2 𝜃 = csc 2 𝜃
tan2 𝜃 + 1 = sec 2 𝜃
Negative Relations:
a
c
90-θ
sin(−𝜃) = −sin 𝜃
θ
b
csc(−𝜃) = −csc 𝜃
tan(−𝜃) = −tan 𝜃
cot(−𝜃) = −cot 𝜃
cos(−𝜃) = cos 𝜃
𝒄𝟐 = 𝒂𝟐 + 𝒃𝟐
Six Trigonometric Functions
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 𝒂
sin 𝜃 =
=
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝒄
sec(−𝜃) = sec 𝜃
Addition and Subtraction Formulas:
sin(𝛼 ± 𝛽) = sin 𝛼 cos 𝛽 ± cos 𝛼 sin 𝛽
cos(𝛼 ± 𝛽) = cos 𝛼 cos 𝛽 ∓ sin 𝛼 sin 𝛽
cos 𝜃 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 𝒃
=
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝒄
tan(𝛼 ± 𝛽) =
tan 𝛼 ± tan 𝛽
1 ∓ tan 𝛼 tan 𝛽
tan 𝜃 =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 𝒂
=
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 𝒃
cot(𝛼 ± 𝛽) =
cot 𝛼 cot 𝛽 ∓ 1
cot 𝛼 ± cot 𝛽
cot 𝜃 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 𝒃
=
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 𝒂
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝒄
sec 𝜃 =
=
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 𝒃
Double Angle Formulas:
sin 2𝜃 = 2 sin 𝜃 cos 𝜃
cos 2𝜃 = 2 cos 2 𝜃 − 1
= cos 2 𝜃 − sin2 𝜃
prepared by: engr.apn
= 1 − 2sin2 𝜃
An oblique triangle is one which does
2 tan 𝜃
1 − tan2 𝜃
not contain a right angle such that it contains
tan 2𝜃 =
cot 2 𝜃 − 1
cot 2𝜃 =
2 cot 𝜃
Half-Angle Formulas:
𝜃
+ 𝑖𝑓 𝑖𝑠 𝑖𝑛 𝐼 𝑜𝑟 𝐼𝐼 𝑄
𝜃
1 − cos 𝜃
2
sin = ±√
{
}
𝜃
2
2
− 𝑖𝑓 𝑖𝑠 𝑖𝑛 𝐼𝐼𝐼 𝑜𝑟 𝐼𝑉 𝑄
2
𝜃
+ 𝑖𝑓 𝑖𝑠 𝑖𝑛 𝐼 𝑜𝑟 𝐼𝑉 𝑄
𝜃
1 + cos 𝜃
2
cos = ±√
{
}
𝜃
2
2
− 𝑖𝑓 𝑖𝑠 𝑖𝑛 𝐼𝐼 𝑜𝑟 𝐼𝐼𝐼 𝑄
2
𝜃
+ 𝑖𝑓 𝑖𝑠 𝑖𝑛 𝐼 𝑜𝑟 𝐼𝐼𝐼 𝑄
𝜃
1 − cos 𝜃
2
tan = ±√
{
}
𝜃
2
1 + cos 𝜃
− 𝑖𝑓 𝑖𝑠 𝑖𝑛 𝐼𝐼 𝑜𝑟 𝐼𝑉 𝑄
2
𝜃
sin 𝜃
1 − cos 𝜃
tan =
=
= csc 𝜃 − cot 𝜃
2 1 + cos 𝜃
sin 𝜃
Multiple Angle Formulas:
sin 3𝜃 = 3 sin 𝜃 − 4 sin3 𝜃
either two acute angles and one obtuse or three
acute angles.
There are four cases that occur in the
solution of oblique triangles.
Case I:
Given two angles in one side.
Case II:
Given two sides and an angle
opposite to
one of them.
Case III: Given two sides and included angle.
Case IV: Given the three sides.
The above cases can be solved using any of of
the following laws:
c
A
cos 3𝜃 = 4 cos 3 𝜃 − 3 cos 𝜃
tan 3𝜃 =
a
C
b
3 tan 𝜃 − tan3 𝜃
1 − 3tan2 𝜃
sin 4𝜃 = 4 sin 𝜃 cos 𝜃 − 8 sin3 𝜃 cos 𝜃
cos 4𝜃 = 8 cos 4 𝜃 − 8 cos 2 𝜃 + 1
tan 4𝜃 =
B
4 tan 𝜃 − 4 tan3 𝜃
1 − 6tan2 𝜃 + tan4 𝜃
Powers of Trigonometric Function:
1
sin2 𝜃 = (1 − cos 2𝜃)
2
1
sin3 𝜃 = (3 sin 𝜃 − sin 3𝜃)
2
1
cos 2 𝜃 = (1 + cos 2𝜃)
2
1
cos 3 𝜃 = (3 cos 𝜃 + cos 3𝜃)
4
Product Formulas:
1
sin 𝛼 cos 𝛽 = [sin(𝛼 + 𝛽) + sin(𝛼 − 𝛽)]
2
1
cos 𝛼 cos 𝛽 = [cos(𝛼 + 𝛽) + sin(𝛼 − 𝛽)]
2
1
sin 𝛼 sin 𝛽 = [cos(𝛼 − 𝛽) − sin(𝛼 + 𝛽)]
2
Solution of Oblique Triangles:
✓ SINE LAW
In any triangle, the ratio of the
side and the sine of the opposite angle
is constant.
𝑎
𝑏
𝑐
=
=
sin 𝐴 sin 𝐵 sin 𝐶
✓ COSINE LAW
In any triangle, the square of any
side is equal to the sum of the square of
the other two sides less twice the
product of these sides and the cosine of
the included angles
𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴
𝑏 2 = 𝑐 2 + 𝑎2 − 2𝑐𝑎 cos 𝐵
𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 cos 𝐶
Logarithm
The word logarithm was taken from the
two greek words “logus” which means ratio and
prepared by: engr.apn
“arithmus” which means number. It was first
𝒆𝒍𝒏 𝒙 = 𝒙
introduced by John Napier and then by Henry
𝐥𝐧 𝟎 = −∞
Briggs.
In modern mathematics, the logarithm of
Relation of Common
Natural Logarithm
number is the exponent to which the based
𝐥𝐧 𝒙 = 𝟐. 𝟑𝟎𝟐𝟔 𝐥𝐨𝐠 𝒙
must be raised to obtain the number.
N = bx
x = logbN
𝐥𝐨𝐠 𝒙 = 𝟎. 𝟒𝟑𝟒𝟑 𝐥𝐧 𝒙
Types of Logarithm:
1. Napierian
Logarithm
Logarithm
to
The Euler’s Number “e”
𝟏 𝒙
𝒆 = 𝐥𝐢𝐦 (𝟏 + ) = 𝟐. 𝟕𝟏𝟖𝟐 …
𝒙→∞
𝒙
is
the
logarithm whose base is the Euler
number e. It is abbreviated as ln which
means loge and was introduced by John
Napier in 1610. The other name given to
Napierian logarithm is understood as
log10.
➢ No real logarithm for negative numbers
➢ The logarithm of negative numbers are
complex or imaginary
➢ The logarithm of 1 to any base is always
zero
➢ The logarithm of 0 is negative infinity if
the base is greater than 1.
2. Briggsian Logarithm is also known
as Common Logarithm that uses 10
as the base. It was introduced by Henry
Briggs in 1616.The abbreviation log is
➢ The logarithm of 0 is positive infinity if
the base is greater than zero but less
than 1.
Spherical Trigonometry:
understood as log10.
Properties of Common Logarithm:
𝐥𝐨𝐠 𝒙𝒚 = 𝐥𝐨𝐠 𝒙 + 𝐥𝐨𝐠 𝒚
𝒙
𝐥𝐨𝐠 = 𝐥𝐨𝐠 𝒙 − 𝐥𝐨𝐠 𝒚
𝒚
𝐥𝐨𝐠 𝒙𝒏 = 𝒏 𝐥𝐨𝐠 𝒙
𝐥𝐨𝐠 𝒙
𝐥𝐨𝐠 𝒚 𝒙 =
𝐥𝐨𝐠 𝒚
𝐥𝐨𝐠 𝒂 𝒂 = 𝟏
𝐥𝐨𝐠 𝒂 𝟏 = 𝟎
𝒂𝐥𝐨𝐠 𝒂 𝒏 = 𝒏
𝐥𝐨𝐠 𝟎 = −∞
Spherical Trigonometry is the branch of
mathematics
which
focuses
on
the
measurement of triangles on the spheres. It is
principally used in navigation and astronomy.
Properties of Natural Logarithm:
𝐥𝐧 𝒙𝒚 = 𝐥𝐧 𝒙 + 𝐥𝐧 𝒚
𝒙
𝐥𝐧 ( ) = 𝐥𝐧 𝒙 − 𝐥𝐧 𝒚
𝒚
Right Spherical Triangle:
A right spherical triangle is the triangle
𝐥𝐧 𝒙𝒏 = 𝒏 𝐥𝐧 𝒙
𝐥𝐧𝒚 𝒙 =
𝐥𝐧 𝒙
𝐥𝐧 𝒚
on the sphere having at least one interior angle
equal to 900. The formulas of the right spherical
𝐥𝐧 𝐞 = 𝟏
triangle can be derived from Napier’s Rules I
𝐥𝐧 𝟏 = 𝟎
and II.
prepared by: engr.apn
An oblique spherical triangle is a triangle
having no right angle. There are six cases arise
from these triangles.
Rule 1. sin-tan-Ad Rule:
The sine of any middle part is equal to
the product of the tangents of the two adjacent
parts.
Case I:
Given three angles.
Case II: Given three sides.
Case III: Given two angles and included sides.
➢ If “a” is the middle part the “B” and “b”
are the adjacent parts:
sin 𝑎 = tan 𝐵𝑐 tan 𝑏 where: tan 𝐵𝑐 = cot 𝐵
sin 𝑎 = cot 𝐵 tan 𝑏
➢ If “A” is the middle part then “cc” and “b”
are the adjacent parts:
sin 𝐴𝑐 = tan 𝑐 𝑐 tan 𝑏
cos 𝐴 = cot 𝑐 tan 𝑏
𝑤ℎ𝑒𝑟𝑒:
tan 𝑐 𝑐 = cot 𝑐 𝑎𝑛𝑑 sin 𝐴𝑐 = cos 𝐴
Case IV: Given two sides and included angles.
Case V: Given two angles and a side opposite
to
one of them.
Case VI:
Given two sides and an angle
opposite
to one of them.
The above cases can be solved using
sine law, cosine law and tangent law.
Rule 2. sin-cos-op Rule:
The sine of any middle part is equal to
the product of the cosines of the two opposite
parts.
Sine Law:
𝑎
𝑏
𝑐
=
=
sin 𝐴 sin 𝐵 sin 𝐶
➢ If “a” is then “cc” and “Ac” are the
opposite parts.
sin 𝑎 = cos 𝑐 𝑐 cos 𝐴𝑐
= sin 𝑐 sin 𝐴
𝑤ℎ𝑒𝑟𝑒:
cos 𝑐 𝑐 = sin 𝑐 𝑎𝑛𝑑 cos 𝐴𝑐 = sin 𝐴
➢ If “Ac” is then “Bc” and “a” are the
opposite parts.
sin 𝐴𝑐 = cos 𝐵𝑐 cos 𝑎
cos 𝐴 = sin 𝐵 cos 𝑎
𝑤ℎ𝑒𝑟𝑒:
sin 𝐴𝑐 = cos 𝐴 𝑎𝑛𝑑 cos 𝐵𝑐 = sin 𝐵
Oblique Spherical Triangle:
Cosine Law for the Sides:
cos 𝐴 = − cos 𝐵 cos 𝐶 + sin 𝐵 sin 𝐶 cos 𝑎
cos 𝐵 = − cos 𝐴 cos 𝐶 + sin 𝐴 sin 𝐶 cos 𝑏
cos 𝐶 = − cos 𝐴 cos 𝐵 + sin 𝐴 sin 𝐵 cos 𝑐
Cosine Law for the Sides:
cos 𝑎 = cos 𝑎 cos 𝑐 + sin 𝑏 sin 𝑐 cos 𝐴
cos 𝑏 = cos 𝑎 cos 𝑐 + sin 𝑎 sin 𝑐 cos 𝐵
cos 𝑐 = cos 𝑎 cos 𝑏 + sin 𝑎 sin 𝑏 cos 𝐶
Note: the sum of interior angles
180° < 𝐴 + 𝐵 + 𝐶 < 540°
Tangent Law:
prepared by: engr.apn
𝐴+𝐵
𝑎+𝑏
) tan (
)
2
2
=
𝐴−𝐵
𝑎−𝑏
tan (
) tan (
)
2
2
tan (
Napier’s Analogies:
𝐴−𝐵
𝑎−𝑏
) tan (
)
2
2
=
𝑐
𝐴+𝐵
tan ( )
sin (
)
2
2
𝑎−𝑏
𝐴−𝐵
sin (
) tan (
)
2
2
=
𝐶
𝑎+𝑏
tan ( )
sin (
)
2
2
𝑎+𝑏+𝑐
2
𝐴+𝐵+𝐶
𝑆=
2
Area of Spherical Triangle:
𝑠=
sin (
𝐴
sin(𝑠 − 𝑏) sin(𝑠 − 𝑐)
sin = √
2
sin 𝑏 sin 𝑐
𝐴=
𝝅𝑹𝟐 𝑬
𝟏𝟖𝟎
𝑤ℎ𝑒𝑟𝑒:
𝐸 = 𝑠𝑝ℎ𝑒𝑟𝑖𝑐𝑎𝑙 𝑒𝑥𝑐𝑒𝑠𝑠
𝐸 = (∠𝐴 + ∠𝐵 + ∠𝐶) − 180°
Spherical Defect:
𝑑 = 360° − (𝑎 + 𝑏 + 𝑐)
THE TERRESTRIAL SPHERE (EARTH)
𝐵
sin(𝑠 − 𝑎) sin(𝑠 − 𝑐)
sin = √
2
sin 𝑎 sin 𝑐
sin
𝐶
sin(𝑠 − 𝑎) sin(𝑠 − 𝑏)
=√
2
sin 𝑎 sin 𝑏
Latitude or Parallel:
These are small circles parallel to the
equator. These will serve as the angular
elevation above or below the equator. The
North Pole is 900 above the equator and South
𝐴−𝐵
𝑎+𝑏
) tan (
)
2
2
=
𝑐
𝐴+𝐵
tan ( )
cos (
)
2
2
cos (
𝑎−𝑏
𝐴+𝐵
) tan (
)
2
2
=
𝐶
𝑎+𝑏
tan ( )
cos (
)
2
2
cos (
cos
𝐴
sin 𝑠 sin(𝑠 − 𝑐)
=√
2
sin 𝑏 sin 𝑐
Pole is 900 below the equator.
Longitude or Meridians:
These are semicircles that run from the
North and South Poles and used to locate how
far east or west from Greenwich, England.
Prime Meridian:
The semi-circle running from the North
to South Pole through Greenwich London.
𝐵
sin 𝑠 sin(𝑠 − 𝑐)
cos = √
2
sin 𝑎 sin 𝑐
cos
𝐶
sin 𝑠 sin(𝑠 − 𝑐)
=√
2
sin 𝑎 sin 𝑏
Opposite the prime meridian is the international
Dateline (IDL).
International Dateline:
It is an arbitrary line established at
cos
𝑎
cos(𝑆 − 𝐵) cos(𝑆 − 𝐶)
=√
2
sin 𝐵 sin 𝐶
𝑏
cos(𝑆 − 𝐴) cos(𝑆 − 𝐶)
cos = √
2
sin 𝐴 sin 𝐶
about 1800 meridian or exactly opposite the
prime
meridian.
The
dateline
does
not
necessarily follow the meridian (semi-circle)
precisely because it is actually zigzags in order
to avoid land masses and archipelagoes.
cos
𝑐
cos(𝑆 − 𝐴) cos(𝑆 − 𝐵)
=√
2
sin 𝐴 sin 𝐵
Bearings:
𝒘𝒉𝒆𝒓𝒆:
prepared by: engr.apn
Measurements from North to South, clockwise
or counterclockwise. It is quadrantal in nature
such that a bearing should never exceed 90 0.
N
N 500 E
N 500 W
550
500
W
E
300
S 500 E
S
Azimuths:
These are clockwise angles usually
measured from a meridian line thus azimuths
used either north or south as their reference.
N
500
W
1500
3050
E
S
prepared by: engr.apn