Trigonometry “Chance favors y mind” Louis Pasteur (1822 - 1895)
Trigonometry
1.1 Plane Angle & Angle Measurements Measurements 1.2 Solution to Right Triangles 1.3 The Six Trigonometric Functions 1.4 Solution to Oblique Triangles . 1.6 Trigonometric Identitie Identities s 1.7 Inverse Trigonometric Functions 1.8 Spherica Sphericall Trigonometry
PLANE ANGLE ANGLE & ANGLE MEASUREMENTS A plane angle is determined by rotating a ray (half-line) about its endpoint called vertex .
Terminal Side
Conversion Factors: ANGLE
1 revolution
= 360 degrees = 2π radians = 400 gradians
VERTEX
Initial Side
= 6400 mils
Types of Angles
Q-1 The measure counterclockwise is
of
2.25
revolutions
A. -835º
C. -810º
B. 805º
D. 810º Conversion Factors: 1 revolution
= 360 degrees = 2π radians = 400 gradians = 6400 mils
4800 mils Q-2 __________degrees.
is
equivalent
A. 135
C. 235
B. 270
D. 142
Conversion Factors: 1 revolution
= 360 degrees = 2π radians = 400 gradians = 6400 mils
to
Q-3 An angular unit equivalent to 1/400 of the circumference of a circle is called: A. degree
C. radian
B. mil
D. grad Conversion Factors: 1 revolution
= 360 degrees = 2π radians = 400 gradians = 6400 mils
Angle Pairs Complementary Angles ∠ A + ∠B = 90 Supplementary Angles ∠ A + ∠B = 180 Explementary Angles
∠ A + ∠B = 360
Q-4 Find the complement of the angle whose supplement is 152º. A. 28º
C. 118º
B. 62º
D. 38º
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
TRIANGLES
Right Triangles
The Pythagorean Theorem:
“ , of the hypotenuse is equal to the sum of the squares of the lengths of the legs”
c2 = a2 + b2
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
c2
>
a2
+b →
c2
<
a2
+ b → The
2
2
The triangle is obtuse triangle is acute
Trigonometric Functions
sin θ =
opposite hypotenuse
=
o h
c ot θ =
adjacent opposite
cos θ =
adjacent hypotenuse
=
a h
sec θ =
hypotenuse adjacent
tan θ =
adjacent
SOHCAH-- TOA TOA SOH-CAH
=
a
csc θ =
opposite
=
=
=
a o h a o
Q-6 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. median =
1 2
2
2
2side + 2side − opposite
2
A. 120 m
C. 122 m
B. 130 m
D. 135 m
ALtitude – perpendicular to opposite side (Intersection: ORTHOCENTER) Angle Bisector – bisects angle
(Intersection: INCENTER)
Median – vertex to midpoint of opposite side (Intersection: CENTROID)
Q-7 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? A. 59.7
C. 69.3
B. 28.5
D. 47.6
Q-8 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.
A. 10
C. 25
B. 15
D. 20
Q-9 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] . 76.31 m
. 73.16 m
B. 73.31 m
D. 73.61 m
Obliliqu Ob que e Tri rian angl gles es The Sine Law
a
=
b
=
c
When to use Sine Law:
• Given two angles and any side. • Given Given two sides and and an angle angle opposite opposite one of them them .
The Cosine Law S ta tan da dard Form : a
2
2
2
= b + c − 2bc CosA
b2
=
a2
+ c − 2ac CosB
c2
=
a2
+ b − 2ab cos C
2
2
Alternative Form : cos A
=
b2
2
+c −a
2
2bc a2 + c 2 − b2 cos B = 2ac a 2 + b2 − c 2 cos C = 2ab
Use the Laws of Cosine if:
Given three sides Given two sides and their included angle
Q-10 In a triangle, find the side c if angle C = 100 , side b = 20 and side a = 15 °
. B. 2 7
. D. 26
Q-11 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
Q-12 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. °
.
.5 m
B. 7.58 m
°
. 47.
m
D. 30.57 m
Area of Triangles
Q-13 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:
. .
54 sq. u.
B. 3.0654 sq. u.
. .
54 sq. u.
D.4.0654 sq. u.
Q-14 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 .
.
Q-15 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
Q-16 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
Q-17 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. . B. 20
. 4 D. 15
R e c ip r o c a l r e la t io n :
s in u
1 csc u
=
co s u
1 se c u
=
Trigonometric Identities
ta n u
Q u o t ie n t r e la t io n
ta n u
s in u co s u
=
P y t h a g o r e a n r e la t io n
A d d it io n &
s in
2
u
co s
+
2
u
=
1
s u b t r a c t io n f o r m u la s i n (u
c o s (u t a n (u
v )
±
v )
± ±
v )
s in u c o s v
=
=
=
co s u co s v
c o s u s in v m
ta n u ± ta n v 1 m ta n u ta n v
D o u b le A n g le fo r m u la :
s in 2 u
=
2 s in u c o s u
co s 2u
=
2 co s
ta n 2 u
=
±
2
u
2 ta n u 1 − ta n 2 u
−
1
s in u s in v
=
1 co tu
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
Q-18 If sec (2x-3) = 1 / sin (5x-9), determine the angle x in degrees
A. 12.56 deg
C. 18.57 deg
. 1 .57 eg
. 10.18 eg
COFUNCTION RELATIONS sin θ = cos ( 90 − θ ) o
o
cos θ = sin(90 − θ ) tan θ = cot ( 90 − θ ) o
sec θ = csc ( 90 − θ ) o
csc θ = sec ( 90 − θ ) o
SOLUTION:
sec2 A = 1
=
1 sin13 A 1
cos 2 A sin13 A sin13 A = cos 2 A
cofunction sin13 A = sin ( 90 − 2 A ) 13 A = 90 − 2 A
A = 6
Q-19 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
sin 2θ = 2 sin θ cos θ tan 2θ =
2tan θ 2
1 − tan θ 2
2
2
2
cos 2θ = cos θ − sin θ = 1 − 2 sin θ = 2 cos θ − 1 4 cos ysin y(1 − 2 sin
2
y)
=
2 ( 2 sin ycos y) (1 − 2 sin
=
2sin 2 y cos 2 y
= sin4 y
2
y)
Q-20 ECE Board Nov. 1996: If sin A = 2.511x , cos A = 3.06x and sin 2A = 3.939x , find the value of x?
.
.
B. 0.256
.
.
D. 0.625
Q-21 Solve for x if tan 3x = 5 tanx
A. 20.705
C. 15.705
B. 30.705
D. 35.705
°
°
°
°
3
tan3θ =
3 tan θ − tan θ 2
1 − 3 tan θ tan 3 x = 5 tan x 3
3 tan x − tan x 2
1 − 3 tan x 3 tan x− tan 2
3
= 5tan x
x= 5 tan x− 15 tan 3 x
14 tan x = 2 tan x 2
tan x =
2
14 x = 20.705
Q-22 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
arctan 2 x + arctan 3 x = 45
let , tan A
=
2 x ; tan B = 3 x
SUBSTITUTE arctan ( tan A ) + arctan ( tan B ) = 45
A + B
=
45
tan ( A + B = 45 ) tan ( A + B ) = tan 45 tan A + tan B
=
1 − tan A tan B SUBSTITUTE 2 x + 3 x 1 − ( 2 x )( 3 x )
=1
2
6 x + 5 x − 1 = 0
x
=
0.1666 & − 1
tan 45
Spherical Trigonometry The study of properties of spherical triangles and their measurements. Conversion Factors 1m nute o arc = 1naut ca m e 1 nautical mile = 6080 ft. 1 nautical mile = 1.1516 statue mile 1 statue mile = 5280 ft. 1 knot = 1 nautical mile per hour
The Terrestrial Sphere
Spherical Triangle A spherical triangle is the triangle enclosed by arcs of three great circles of a sphere. ① Sum of Three vertex angle :
A + B + C > 180° A + B + C < 540° ② Sum of any two sides :
b+c
>
a
a+c
>b
③ Sum of three sides :
0° < a + b
+c <
360 °
④ Spherical Excess :
E = ( A + B + C ) − 180 ° ⑤ Spherical Defect :
D = 360° − ( a + b + c )
SPHERICAL TRIANGLES:
Law of sines: sin a sin
=
sin b sin B
=
sin c sin C
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
Q-23 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 .
.
cos c = cos a cos b + sin a sin b cos C cos ( 80 ) = cos ( 50 ) cos b + sin ( 50 ) sin b cos ( 90 ) 0.1736 = 0.6428cos b + 0
b=7 .
Q-24 Given an isosceles triangle with angle A=B=64 degrees, and side b=81 degrees . What is the value of angle C?
.
o
o
B. 135 10 '
.
o
o
D. 150 25'
COSINE LAW FOR ANGLES:
cos A = − cos Bcos C + sin Bsin Ccos a cos B = − cos Acos C + sin Asin Ccos b cos C = − cos A cos B + sin A sin B cos c
Q-24 Given an isosceles triangle with angle A=B=64 degrees, and side b=81 degrees . What is the value of angle C?
.
o
o
B. 135 10 '
.
o
o
D. 150 25'
COSINE LAW FOR ANGLES:
cos A = − cos Bcos C + sin Bsin Ccos a cos B = − cos Acos C + sin Asin Ccos b cos C = − cos A cos B + sin A sin B cos c cos B = − cos Acos C + sin Asin Ccos b cos 64 = − cos(64) cos C + sin 64 sin C cos 81 0.4384 = −0.4384cos C + 0.1406sin C
SUBSTITUTE 0.4384 = −0.4384 cos(144 26 ') + 0.1406 sin (144 26 ' ) o
0.4384 = 0.4384
o
POP QUIZ…
1. Three circles of radii 3,4, and 5 inches, respectively are tangent to each other extremely. Find the largest angle of a triangle formed by joining the centers.
A.
72.6 deg
B.
75.1 deg
C.
73.4 deg
D.
73.3 deg
2. If sinA = 4/5 and sinB = 7/25, what is sin(A+B) if A is in the 3 rd quadrant and B is in the 2nd quadrant?
A.
-3/5
B.
4/5
C.
3/5
D.
2/5
3. Find the value of (1 + i)12 .
A.
64
B.
-64
. D.
-64i
4. The altitude of the sides of a triangle intersect at the point known as
A.
Incenter
. C.
Orthocenter
D.
Centroid
5. A triangle inscribed in a given triangle whose vertices are the feet of the three perpendiculars to the sides from the same point inside the given triangle.
A.
Pedal triangle
B.
Scalene triangle
C.
Escribed triangle
D.
Egyptian triangle
6. The angle of inclination of ascend of a road having 8.25 % grade is ____ degrees.
A. .
4.72 .
C.
5.12
D.
1.86
7. The sides of a triangle are 8, 10, and 14. Determine the radius of the inscribed circle.
A.
18.9
B.
19.8
C.
17.9
D.
16.9
8. From the top of the 100-ft-tall building a man observes a car moving toward the building. If the angle of depression of the car changes from 22 to 46 during the period of observation, how far does the car travel? °
A. 120 B. 151
°
C. 171 D. 180
9. Calculate the angle of elevation of the line of sight of a person whose eye is 1.7 m above the ground, and is looking at the top of a tree which is 27.5 m away on level ground and 18.6 m high. A. 30 degrees B. 12 degrees C. 25 degrees D. 32 degrees
10. If z varies directly as x and inversely as y, find the percentage change in z if x increases by 20% and y increases by 25%. A. z decreases by 5% . C. z decreases by 4% D. z decreases by 6.25%
Check Time!!!
1. Three circles of radii 3,4, and 5 inches, respectively are tangent to each other extremely. Find the largest angle of a triangle formed by joining the centers.
A.
72.6 deg
B.
75.1 deg
C.
73.4 deg
D.
73.3 deg
2. If sinA = 4/5 and sinB = 7/25, what is sin(A+B) if A is in the 3 rd quadrant and B is in the 2nd quadrant?
A.
-3/5
B.
4/5
C.
3/5
D.
2/5
5. A triangle inscribed in a given triangle whose vertices are the feet of the three perpendiculars to the sides from the same point inside the given triangle.
A.
Pedal triangle
B.
Scalene triangle
C.
Escribed triangle
D.
Egyptian triangle
6. The angle of inclination of ascend of a road having 8.25 % grade is ____ degrees.
A. .
4.72 .
C.
5.12
D.
1.86
7. The sides of a triangle are 8, 10, and 14. Determine the radius of the inscribed circle.
A.
18.9
B.
19.8
C.
17.9
D.
16.9
8. From the top of the 100-ft-tall building a man observes a car moving toward the building. If the angle of depression of the car changes from 22 to 46 during the period of observation, how far does the car °
A. 120 B. 151
°
C. 171 D. 180
9. Calculate the angle of elevation of the line of sight of a person whose eye is 1.7 m above the ground, and is looking at the top of a tree which is 27.5 m away on level ground and 18.6 m high. A. 30 degrees B. 12 degrees C. 25 degrees D. 32 degrees
