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vertex
Angle
NOTES IN TRIGONOMETRY
Angles and its Measure:
Angle is the space between intersecting
rays or lines. The point of intersection is called
vertex. Angles are positive when measured
counterclockwise (ccw) and negative in
clockwise (cw) direction.
Units Used in Measuring Angles:
1 revolution = 360 degrees
= 2Π radian
= 400 grads
= 400 gons
= 6400 mils
Names
Angle Equivalent
in Degrees
zero angle θ= 0
0
acute angle 0
0
< θ< 90
0
right angle θ= 90
0
obtuse angle 0
0
< θ< 180
0
straight angle θ= 180
0
Reflex angle 180
0
< θ< 360
0
full angle or Perigon θ= 360
0
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
used to some problems:
Complement of A = 90°- A
Supplement of B = 180°- B
Explement of C = 360°- C
Classifications of Triangles:
Triangles are classified in terms of their
interior angles and the sides.
➢ For angles as reference:
o Right triangle - triangle with one
interior angle equal to 90
0
o Oblique triangle - triangles are
classified into:
▪ Acute Triangle - triangles
with one interior angle
equal to 90°.
▪ Obtuse Triangle -
triangles in which one of
the interior angle is more
than 90 °but less than
0
▪ Equiangular Triangle -
triangles in which all of
the interior angle are
equal.
➢ For sides as reference:
o Isosceles Triangle - triangle with
two sides equal.
o Scalene Triangle - triangle with
none of the sides are equal.
o Equilateral Triangle - triangle
with all sides are equal. Also
called equiangular Triangle.
Schwartz’s Inequality:
The sum of any two sides of any triangle
is greater than the third side.
Angle of Depression and Elevation:
A C
B
c
a
b
θ
90 - θ
a
c
b
The angle of depression is the angle
from horizontal down to the line of sight from
the observer to an object below. The angle of
elevation is the angle from the horizontal up
to the line of sight from the observer to an
object above. The angle of elevation is equal to
the angle of depression.
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.
𝟐
𝟐
𝟐
Six Trigonometric Functions
sin 𝜃 =
cos 𝜃 =
tan 𝜃 =
cot 𝜃 =
sec 𝜃 =
csc 𝜃 =
TRIGONOMETRIC IDENTITIES
Co-function Relations:
sin( 90 − 𝜃) = cos 𝜃
cos
= sin 𝜃
tan( 90 − 𝜃) = cot 𝜃
cot( 90 − 𝜃) = tan 𝜃
sec( 90 − 𝜃) = csc 𝜃
csc
= sec 𝜃
Reciprocal Relations:
sin 𝜃 =
csc 𝜃
cot 𝜃 =
tan 𝜃
cos 𝜃 =
sec 𝜃
sec 𝜃 =
cos 𝜃
tan 𝜃 =
cot 𝜃
csc 𝜃 =
sin 𝜃
Tangent and Co-tangent Relations:
tan 𝜃 =
sin 𝜃
cos 𝜃
cot 𝜃 =
cos 𝜃
s 𝜃
Pythagorean Relations:
sin
2
𝜃 + cos
2
1 + cot
2
𝜃 = csc
2
tan
2
𝜃 + 1 = sec
2
Negative Relations:
sin(−𝜃) = −sin 𝜃
csc(−𝜃) = −csc 𝜃
tan
= −tan 𝜃
cot(−𝜃) = −cot 𝜃
cos(−𝜃) = cos 𝜃
sec
= sec 𝜃
Addition and Subtraction Formulas:
sin
= sin 𝛼 cos 𝛽 ± cos 𝛼 sin 𝛽
cos(𝛼 ± 𝛽) = cos 𝛼 cos 𝛽 ∓ sin 𝛼 sin 𝛽
tan
tan 𝛼 ± tan 𝛽
1 ∓ tan 𝛼 tan 𝛽
cot
cot 𝛼 cot 𝛽 ∓ 1
cot 𝛼 ± cot 𝛽
Double Angle Formulas:
sin 2 𝜃 = 2 sin 𝜃 cos 𝜃
cos 2 𝜃 = 2 cos
2
= cos
2
𝜃 − sin
2
“arithmus” which means number. It was first
introduced by John Napier and then by Henry
Briggs.
In modern mathematics, the logarithm of
number is the exponent to which the based
must be raised to obtain the number.
N = b
x
x = log b
N
Types of Logarithm:
- Napierian Logarithm is the
logarithm whose base is the Euler
number e. It is abbreviated as ln which
means log e
and was introduced by John
Napier in 1610. The other name given to
Napierian logarithm is understood as
log 10
- 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
understood as log 10
Properties of Common Logarithm:
𝒏
𝒚
𝒂
𝒂
𝐥𝐨𝐠 𝒂
𝒏
Properties of Natural Logarithm:
𝒏
𝒚
𝒍𝒏 𝒙
Relation of Common Logarithm to
Natural Logarithm
The Euler’s Number “e”
𝒙→∞
𝒙
➢ 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.
➢ The logarithm of 0 is positive infinity if
the base is greater than zero but less
than 1.
Spherical Trigonometry:
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.
Right Spherical Triangle:
A right spherical triangle is the triangle
on the sphere having at least one interior angle
equal to 90
0
. The formulas of the right spherical
triangle can be derived from Napier’s Rules I
and II.
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.
➢ If “a” is the middle part the “B” and “b”
are the adjacent parts:
sin 𝑎 = tan 𝐵
𝑐
tan 𝑏 where: tan 𝐵
𝑐
= cot 𝐵
sin 𝑎 = cot 𝐵 tan 𝑏
➢ If “A” is the middle part then “c
c
” and “b”
are the adjacent parts:
sin 𝐴
𝑐
= tan 𝑐
𝑐
tan 𝑏
cos 𝐴 = cot 𝑐 tan 𝑏
tan 𝑐
𝑐
= cot 𝑐 𝑎𝑛𝑑 sin 𝐴
𝑐
= cos 𝐴
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.
➢ If “a” is then “c
c
” and “A
c
” are the
opposite parts.
sin 𝑎 = cos 𝑐
𝑐
cos 𝐴
𝑐
= sin 𝑐 sin 𝐴
cos 𝑐
𝑐
= sin 𝑐 𝑎𝑛𝑑 cos 𝐴
𝑐
= sin 𝐴
➢ If “A
c
” is then “B
c
” and “a” are the
opposite parts.
sin 𝐴
𝑐
= cos 𝐵
𝑐
cos 𝑎
cos 𝐴 = sin 𝐵 cos 𝑎
sin 𝐴
𝑐
= cos 𝐴 𝑎𝑛𝑑 cos 𝐵
𝑐
= sin 𝐵
Oblique Spherical Triangle:
An oblique spherical triangle is a triangle
having no right angle. There are six cases arise
from these triangles.
Case I: Given three angles.
Case II: Given three sides.
Case III: Given two angles and included sides.
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.
Sine Law:
sin 𝐴
sin 𝐵
sin 𝐶
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
Tangent Law:
Measurements from North to South, clockwise
or counterclockwise. It is quadrantal in nature
such that a bearing should never exceed 90
0
Azimuths:
These are clockwise angles usually
measured from a meridian line thus azimuths
used either north or south as their reference.
N
S 50
0
E
N 50
0
W
N 50
0
E
S
E
W
55
0
30
0
50
0
150
0
305
0
N
S
E
W
50
0
