This document provides an overview of trigonometry including plane and spherical trigonometry. It covers topics such as angle measurements, the six trigonometric functions, solving right triangles using the Pythagorean theorem, solving oblique triangles using laws of sines and cosines, inverse trigonometric functions, trigonometric identities, and area of triangles. It also includes sample problems and their solutions related to these topics.

1. Trigonometry
Ericson D. Dimaunahan, ECE
“Chance favors
only the prepared
mind”
Louis Pasteur (1822 - 1895)

2. 1.1 Plane Angle & Angle Measurements
1.2 Solution to Right Triangles
1.3 The Six Trigonometric Functions
1.4 Solution to Oblique Triangles
1.5 Area of Triangles
1.6 Trigonometric Identities
1.7 Inverse Trigonometric Functions
1.8 Spherical Trigonometry
Trigonometry

3. Plane Angle & Angle Measurements
A plane angle is determined by rotating a ray (half-line)
about its endpoint called vertex.
Conversion Factors:
1 revolution = 360 degrees
= 2π radians
= 400 gradians
= 6400 mils

6. Types of Angles

11. Q-1 The measure of 2.25 revolutions
counterclockwise is
A. -835º C. -810º
B. 805º D. 810º
Conversion Factors:
1 revolution = 360 degrees
= 2π radians
= 400 gradians
= 6400 mils

12. Q-2 4800 mils is equivalent to
__________degrees.
A. 135 C. 235
B. 270 D. 142
Conversion Factors:
1 revolution = 360 degrees
= 2π radians
= 400 gradians
= 6400 mils

13. A. degree C. radian
B. mil D. grad
Q-3 An angular unit equivalent to 1/400 of the
circumference of a circle is called:
Conversion Factors:
1 revolution = 360 degrees
= 2π radians
= 400 gradians
= 6400 mils

17. Q-4 Find the complement of the angle whose
supplement is 152º.
A. 28º C. 118º
B. 62º D. 38º

18. Q-5 A certain angle has an explement 5 times
the supplement. Find the angle. [ECE Board
Nov.2002]
A. 67.5 degrees C. 135 degrees
B. 108 degrees D. 58.5 degrees

19. Q-6 What is the reference angle and one
coterminal angle , respectively of 135º.
A. -45º, -225º
B. -45º, 225º
C. 45º, 225º
D. 45º, -225º
RELATED ANGLES – angles
that have the same absolute
values for their trigonometric
functions. The acute angle is the
reference angle.
Ex. 20, 160, 200, 340

23. Right Triangles
The Pythagorean Theorem:
“In a right triangle, the square of the length
of the hypotenuse is equal to the sum of
the squares of the lengths of the legs”
c2 = a2 + b2

24. Note:
In any triangle, the sum of any two sides must be
greater than the third side; otherwise no triangle can
be formed.
If, 2 2 2
2 2 2
2 2 2
c a b The triangle is right
c a b The triangle is obtuse
c a b The triangle is acute
  
  
  

25. Trigonometric Functions
     
     
     
opposite o adjacent a
sin cot
hypotenuse h opposite o
adjacent a hypotenuse h
cos sec
hypotenuse h adjacent a
opposite o hypotenuse h
tan csc
adjacent a opposite o
SOH-CAH-TOA

26.   
opposite o
sin
hypotenuse h
Sine Functions

27. Q-7 The sides of a triangular lot are130 m,
180 m and 190 m. This lot is to be divided by a
line bisecting the longest side and drawn from
the opposite vertex. Find the length of this line.
A. 120 m C. 122 m
B. 130 m D. 125 m
Altitude – perpendicular to opposite side (Intersection: ORTHOCENTER)
Angle Bisector – bisects angle (Intersection: INCENTER)
Median – vertex to midpoint of opposite side (Intersection: CENTROID)
2 2 2
1
2 2
2
median side side opposite
  

29. A. 59.7 C. 69.3
B. 28.5 D. 47.6
Q-8 The angle of elevation of the top of the
tower from a point 40 m. from its base is the
complement of the angle of elevation of the
same tower at a point 120 m. from it. What is
the height of the tower?
 90 



30. A. 10 C. 25
B. 15 D. 20
Q-9 One leg of a right triangle is 20 cm and
the hypotenuse is 10 cm longer that the other
leg. Find the length of the hypotenuse.

31. A. 76.31 m C. 73.16 m
B. 73.31 m D. 73.61 m
Q-10 A man finds the angle of elevation of the
top of a tower to be 30 degrees. He walks 85
m nearer the tower and finds its angle of
elevation to be 60 degrees. What is the height
of the tower ? [ECE Board Apr. 1998]
30 60
30

32. Oblique Triangles
a b c
sinA sinB sinC
 
The Sine Law
When to use Sine Law:
• Given two angles and any side.
• Given two sides and an angle opposite one of them .

33. 2 2 2
2 2 2
2 2 2
2 2 2
2 2 2
2 2 2
Standard Form : Alternative Form :
b c a
a b c 2bcCosA cosA
2bc
a c b
b a c 2acCosB cosB
2ac
a b c
c b c 2bccosC cosC
2ab
 
   
 
   
 
   
The Cosine Law
Use the Laws of Cosine if:
Given three sides
Given two sides and their included angle

34. Q-11 In a triangle, find the side c if angle C
= 100 , side b = 20 and side a = 15
A. 28 C. 29
B. 27 D. 26

35. Q-12 Points A and B 1000 m apart are plotted on
a straight highway running east and west. From
A , the bearing of a tower C is 32 degrees W of N
and from B the bearing of C is 26 degrees N of
E . Approximate the shortest distance of tower C
to the highway. [ECE Board Apr. 1998:]
A. 364 m C. 394 m
B. 374 m D. 384 m

36. Q-13 A PLDT tower and a monument stand on a
level plane . The angles of depression of the top
and bottom of the monument viewed from the top
of the PLDT tower are 13 and 35 respectively.
The height of the tower is 50 m. Find the height
of the monument.
A. 33.51 m C. 47.30 m
B. 7.58 m D. 30.57 m
13
35
50

37. Area of Triangles

39. Q-14 Given a right triangle ABC. Angle C is the
right angle. BC = 4 and the altitude to the
hypotenuse is 1 unit. Find the area of the
triangle. ECE Board Apr.2001:
A. 2.0654 sq. u. C.1.0654 sq. u.
B. 3.0654 sq. u. D.4.0654 sq. u.

40. Q-15 In a given triangle ABC, the angle C is
34°, side a is 29 cm, and side b is 40 cm.
Solve for the area of the triangle.
A. 324.332 cm2 C. 317.15 cm2
B. 344.146 cm2 D. 343.44 cm2

41. Q-16 A right triangle is inscribed in a circle such
that one side of the triangle is the diameter of a
circle. If one of the acute angles of the triangle
measures 60 degrees and the side opposite that
angle has length 15, what is the area of the
circle? ECE Board Nov. 2002
A. 175.15 C. 235.62
B. 223.73 D. 228.61

42. Q-17 The sides of a triangle are 8 cm , 10 cm,
and 14 cm. Determine the radius of the
inscribed and circumscribing circle.
A. 3.45, 7.14 C. 2.45, 8.14
B. 2.45, 7.14 D. 3.45, 8.14

43. Q-18 Two triangles have equal bases. The altitude
of one triangle is 3 cm more than its base while
the altitude of the other is 3 cm less than its base.
Find the length of the longer altitude if the areas of
the triangle differ by 21 square centimeters.
A. 10 C. 14
B. 20 D. 15

44. Trigonometric Identities
 
 
 
2 2
Reciprocal relation :
1 1 1
sinu cosu tanu
csc u sec u cot u
Quotient relation
sinu
tanu
cosu
Pythagorean relation
sin u cos u 1
Addition & subtraction formula
sin u v sinucos v cosu sin v
cos u v cosucos v sinu sin v
tan u v
  

 
  
 
 
2
2
tanu tan v
1 tanu tan v
Double Angle formula :
sin 2u 2 sinucosu
cos 2u 2cos u 1
2 tanu
tan 2u
1 tan u


 



45. Inverse Trigonometric Functions
The Inverse Sine Function
y = arc sin x iff sin y = x
The Inverse Cosine Function
y = arc cos x iff cos y = x
The Inverse Tangent Function
y = arc tan x iff tan y = x

46. Q-19 If sec 2A = 1 / sin 13A, determine the
angle A in degrees
A. 5 degrees C. 3 degrees
B. 6 degrees D. 7 degrees

47.  
 
 
 
sin cos 90
cos sin(90 )
tan cot 90
sec csc 90
csc sec 90
 
 
 
 
 
 
 
 
 
 
COFUNCTION RELATIONS
 
1
sec2
sin13
1 1
cos2 sin13
sin13 cos2
sin13 sin 90 2
13 90 2
6
A
A
A A
A A
cofunction
A A
A A
A



 
 

SOLUTION:

48. Q-20 ECE Board Nov.2003
Simplify the expression
4 cos y sin y (1 – 2 sin2y).
A. sec 2y C. tan 4y
B. cos 2y D. sin 4y

49. 2
2 2 2 2
sin 2 2sin cos
2tan
tan 2
1 tan
cos2 cos sin 1 2sin 2cos 1
  



    



     
 
  
2
2
4cos sin 1 2sin
2 2sin cos 1 2sin
2sin 2 cos2
sin 4
y y y
y y y
y y
y

 



50. Q-21 ECE Board Nov. 1996:
If sin A = 2.511x , cos A = 3.06x and sin 2A
= 3.939x , find the value of x?
A. 0.265 C. 0.562
B. 0.256 D. 0.625

51. Q-22
Solve for x if tan 3x = 5 tanx
A. 20.705 C. 15.705
B. 30.705 D. 35.705

52. 3
2
3
2
3 3
2
2
3tan tan
tan3
1 3tan
tan3 5tan
3tan tan
5tan
1 3tan
3tan tan 5tan 15tan
14tan 2tan
2
tan
14
20.705
x x
x x
x
x
x x x x
x x
x
x
 









  




53. Q-23 If arctan2x + arctan3x = 45 degrees,
what is the value of x?
ECE Nov. 2003
A. 1/6 C.1/5
B. 1/3 D.1/4

54.    
 
 
  
2
arctan 2 arctan 3 45
, tan 2 ;tan 3
arctan tan arctan tan 45
45
tan 45
tan tan 45
tan tan
tan 45
1 tan tan
2 3
1
1 2 3
6 5 1 0
0.1666 & 1
x x
let A x B x
SUBSTITUTE
A B
A B
A B
A B
A B
A B
SUBSTITUTE
x x
x x
x x
x
 
 
 
 
 
 






  
 

55. Spherical Trigonometry
The study of properties of spherical triangles
and their measurements.
The Terrestrial
Sphere
1minute of arc 1nautical mile
1nautical mile 6080 ft.
1nautical mile 1.1516 statue mile
1statue mile 5280 ft.
1knot 1nautical mile per hour





Conversion Factors

56. Spherical Triangle
A spherical triangle is the triangle enclosed by arcs of three great
circles of a sphere.
 
 
Sum of Three vertex angle :
A B C 180
A B C 540
Sum of any two sides :
b c a
a c b
a b c
Sum of three sides :
0 a b c 360
Spherical Excess :
E A B C 180
Spherical Defect :
D 360 a b c
   
   
 
 
 
     
    
    
①
②
③
④
⑤

57. sin sin sin
sin sin sin
a b c
A B C
 
cos cos cos sin sin cos
cos cos cos sin sin cos
cos cos cos sin sin cos
a b c b c A
b a c a c B
c a b a b C
 
 
 
SPHERICAL TRIANGLES:
Law of sines:
Law of cosines (FOR SIDES):

58. Q-26 A spherical triangle ABC has sides a =
50°, c = 80°, and an angle C = 90°. Find
the third side “b” of the triangle in degrees.
A. 75.33 degrees C. 74.33 degrees
B. 77.25 degrees D. 73.44 degrees

59.        
cos cos cos sin sin cos
cos 80 cos 50 cos sin 50 sin cos 90
0.1736 0.6
74.3
428cos
3
0
c a b a b C
b
b b
b
 
 

 

60. Q-27 Given an isosceles triangle with angle
A=B=64 degrees, and side b=81 degrees .
What is the value of angle C?
A. C.
B. D.
144 26'
135 10'
120 15'
150 25'

61. cos cos cos sin sin cos
cos cos cos sin sin cos
cos cos cos sin sin cos
A B C B C a
B A C A C b
C A B A B c
  
  
  
COSINE LAW FOR ANGLES:
 
cos cos cos sin sin cos
cos64 cos(64)cos sin 64sin cos81
0.4384 0.4384cos 0.1406sin
0.4384 0.4384cos(144 26') 0.1406sin 144 26'
0.4384 0.4384
B A C A C b
C C
C C
SUBSTITUTE
  
  
  
  


62. END...