TRIGONOMETRY
1. Trigonometry – is a
branch of mathematics that deals with the relationship between the angles and
sides of a triangle and the theory of the periodic functions connected with
them.
2. Triangle – a figure formed by three line
segments joining three points that are not in the same plane.
3. Right triangle – a triangle with at least one
right angle.
4. Oblique triangle – any triangle which has no
right angle.
Oblique triangles are classified according
to sides and according to angles as :
i) According to sides ii) According to angles
a) equilateral triangle a)
equiangular triangle
b) isosceles triangle b)
acute triangle
c) scalene triangle c) obtuse triangle
5. Congruent triangles – two triangles with the
same shape and size.
6. Similar triangles – two triangles with the
same shape but not necessarily of the same size.
7. Altitude of a
triangle – the length of the line segment from any vertex of a triangle that
is perpendicular to the opposite side.
8. Hypotenuse – the longest side of the triangle
that is opposite to the right angle.
9. Perimeter of a
triangle – is the distance around the triangle and is equal to the sum of the
three sides.
10. The area of a
triangle is ½ of the base ( b ) times
the altitude ( a ).
A = ½ b a
If the three sides are given, the area
is given by the formula
where s = ½
( a + b + c )
RIGHT TRIANGLES
Figure 1
11. Pythagorean
theorem – in any right triangle, the square of the hypotenuse is equal to the
sum of the squares of the other two sides.
c2 = a2 + b2
b2 = c2 – a2
a2 = c2 – b2
* Proofs of Pythagorean theorem
( as presented in the classroom
discussion )
Examples : Find the missing side, the perimeter and the
area of the given right triangle.
1.
a = 12 cm , c =
13 cm 7. a
= 24 cm , c
= 40 cm
2. b
= 12 cm , a = 9 cm 8. b
= 36 cm , c
= 45 cm
3. c = 35 cm, b = 21
cm 9. a
= 33 cm , b
= 44 cm
4. a = 36
cm, b = 48 cm 10.
b = 48 cm ,
c = 80 cm
5. a =
81 cm, c = 135
cm 11. a
= 63 cm , c
= 105 cm
6. c = 12 cm, a
= 6 cm 12. a
= 78 cm ,
b = 104 cm
13.
Will a round glass table top, 2.5 m in diameter, fit through a doorway
which is 2.13 m
high and 0.92 m
wide ?
14. A rectangle has a base of 12 cm. If
the diagonal is 15 cm, determine the altitude of the rectangle.
15.
The area of a square is 128 cm2. Determine the length of its
diagonal.
From Geometry :
The sum of the measure of the
three angles of a triangle is equal to
180°
From the
figure above :
a +
b +
90° = 180°
a +
b = 180° – 90°
a +
b =
90°
Hence, the sum of the measure of
the acute angles of a right triangle is equal to 90°.
Example :
Find a if b
= 39° 28’.
Solution : a
= 90° – 39° 28’ = 50° 32’
Exercises : Find a
or b
if
1. a = 36° 24’ 32’’ 6. b = 56°
43’ 27’’
2. b =
68° 13’ 7. a= 62° 54’ 12’’
3. a =
47° 19’
8. b = 73° 24’ 32’’
4. a =
17° 21’ 36” 9. a = 16° 48’
5. b =
67° 34’ 26” 10. b
= 22° 47’
PROPERTIES OF SIMILAR
TRIANGLES
Basic principle :
The
ratio of any two sides of a right triangle with an acute angle depends only on
the size of the angle and not on the size of the triangle.
Memory aid
SOH
CHO
CAH
SHA
TOA
CAO
The six
trigonometric ratios are called trigonometric functions.
The
value of each of the six trigonometric functions of an acute angle is determined when the acute
angle is given. Furthermore, it can be
shown that, if the value of one of the six trigonometric functions of an acute
angle is equal to the value of the same function of a second acute angle, the
two acute angles are equal.
Example
1. OA = 9 cm,
AX = 12 cm, OX = 15 cm
OB = 21 cm, BY = 28 cm,
OY = 35 cm
OC = 33 cm, CZ = 44 cm,
OZ = 55 cm
2. OA = 5 cm,
AX = 12 cm, OX = 13 cm
OB = 10 cm, BY = 24 cm,
OY = 26 cm
OC = 20 cm, CZ = 48 cm,
OZ = 52 cm
3.
OA = 6 cm, AX = 8 cm, OX =
10 cm
OB = 12 cm, BY = 16 cm,
OY = 20 cm
OC = 24 cm, CZ = 32 cm,
OZ = 40 cm
FUNCTIONS
OF ACUTE ANGLE
1. Sin a 4. csc a
2. cos a 5. sec a
3. tan a
6. Cot a
Exercises
1.
If sin a = 3/5, find cos a, tan a,
sec a, csc a
and cot a.
2.
If tan a = 5/12, find cos a, sin a,
sec a, csc a
and cot a.
3.
If cot a = 8/15, find cos a, tan a,
sec a, csc a
and sin a.
4.
If cos a = 12/13, find sin a, tan a,
sec a, csc a
and cot a.
5.
If csc a = 2, find cos a, tan a,
sec a, sin a
and cot a.
6.
If sec a = 5/4, find cos a, tan a,
sin a, csc a
and cot a.
7.
If cos a = 4/5, find sin a, tan a,
sec a, csc a
and cot a.
8.
If tan a = 5/12, find cos a, sin a,
sec a, csc a
and cot a.
9.
Given a right triangle
Prove that a)
sin a/cos a
= tan a
b) sin a cos a
= ab /( a2 + b2 )
c) ( sin a ) 2 + (
cos a ) 2 = 1
d) ( sec a ) 2 – ( tan a ) 2 = 1
e) ( csc a ) 2 – (
cot a ) 2 = 1
10. In the figure, RS = 60 cm, cos a = 4/5, sin O = 5/13, determine the length of the sides
QR, QS and
QP.
11. Using the same figure, determine the unknowns
if sin a = 3/5, and cos O = 12/13 .
12. Two guy wires are attached to a pole 12 m
above the ground level. They make angles
of 43° and
72° with the ground at points
which are in a line with the base of the pole.
How long are the wires ?
13. At low tide the angle of elevation to the top
of 12 m tall tree from the water’s edge
is
6° 30'. At high tide the angle of elevation to
the top of the tree is 6°. How high does the water level rises during the high tide along
the line perpendicular to the shore ?
14.
Determine the height of a tower if from a
window in a building 95.735 m away the angle
of depression to the base of the tower is 68° while the angle of elevation to the top is 62°.
of depression to the base of the tower is 68° while the angle of elevation to the top is 62°.
15. A light house casts a shadow of 9 m when the
angle of elevation of the sun is 56°. How
high is the light house?
16. The angle of elevation to a balloon from a
point on the ground is 32°. After a
vertical
ascent of
68 m the angle of depression from the balloon to the same point on the
ground is 47°. What are the original and present
heights of the balloon ?
17.
From a point 25 m from the base of a tower, a bird flies to its top in a
straight path at an
angle of elevation of 75° 32’. How long is the flight path of the bird ?
What is the height
of the tower ?
sin a
= a/c cos b = a /c
cos a
= b/c sin b = b /c
tan a
= a /b cot b = a /b
sec a
= c /b csc b = c /b
csc a
= c / a sec b = c / a
cot a
= b / a tan b = b / a
sin a
= cos b
= a / c
cos a
= sin b
= b / c
tan a
= cot b
= a / b
sec a
= csc b
= c / b
csc a
= sec b
= c / a
cot a
= tan b
= b / a
FUNCTIONS
AND CO-FUNCTIONS
Sine ---
cosine
Cosine ----
sine
Tangent --- cotangent
Cotangent ---
tangent
Secant
--- cosecant
Cosecant
--- secant
RECIPROCAL FUNCTIONS
sin
a = 1/ csc a
cos a = 1/
sec a
tan a = 1/
cot a
csc a = 1/
sin a
sec a = 1/ cos
a
cot a = 1/
tan a
Any
function of a is equal
to the co-function of the complement of a ( B ).
Examples :
1. sin
30° = cos 60°
= 0.5
2. tan
15° = cot
75° = 0.26794919
3. sec 36°
= csc 54° = 1.236067977
4. cos
25° = sin
65° = 0.906307787
SOLUTION
OF RIGHT TRIANGLES
Formulas :
1.
a2 + b2 = c2
2. a
+ b
= 90°
3. sin a
= a / c = cos b
4. cos a
= b / c = sin b
5. tan a
= a / b = cot b
6. cot a
= b / a = tan b
7. sec a
= c / b = csc b
8. csc a
= c / a = sec b
Example
:
Find the acute angles of the triangle
whose base is 15 cm and an altitude of 20 cm.
From the figure above, tan a =
20/15 = 1.333333333
a = Arc tan 1.3333333 =
53.13°
tan b = b/a
= 15/ 20 = 0.75
b = Arct
tan 0.75 = 36.87°
Exercises : Find the measure of the angle a and B of the following triangles using
calculator.
1. a =
18, b = 25 6. a =
21, c = 40
2. a = 40,
b = 60 7. b =
50, c = 100
3. b = 17.321,
c = 20 8. a =
12, c = 15
4. a = 12 , b =
24 9. a = 15, b =
28
5. b = 18, c = 30 10. b = 35, c = 44
SUGGESTIONS
FOR SOLVING RIGHT TRIANGLES.
1.
Make a preliminary sketch roughly to scale for the given data.
2. To find any unknown part, use a formula which
involves it but no other unknown.
3. Check the result by substituting it in any of
the equations from 1 to 8.
ANGLE
OF ELEVATION AND DEPRESSION
* Angle of elevation – the angle formed by a
horizontal ray and the observer’s “line
of sight” to any point above the horizontal.
* Angle of depression – the angle formed by a
horizontal ray and the observer’s “line of
sight” to any point below the horizontal.
Examples
:
1. A 120 m high tower casts a shadow of 60 m.
Determine the angle of elevation of the sun.
2. A light house is 14 m high. Determine the
length of its shadow when angle of elevation of the sun is 52°.
3. From an airplane flying at 2,135 m high above
the ground, the angle of depression of a
landing field is 19° 32’. Determine the
line of sight distance and the ground distance from the plane to the field. ( The ground distance
is the distance of the plane from the field to the point on the ground under the plane. )
4. Determine the height of the a tower if the
angle of elevation of its top is 56° 35’
when seen from a point 20.6 m from the base of the tower.
5. In flying upward for
1,152 m along a straight inclined
path, an airplane rises 180 m.
Determine the climbing angle of the plane.
( Climbing angle is the angle of inclination of the plane from the horizontal ).
6. From a ground distance of 1,100 m, an airplane starts a straight glide for the
edge of an airfield at a gliding angle of 16° from the horizontal. From what altitude did the glide starts ?
7. From an observation point on a level ground
the angle of elevation of the base of a tree on a mountain is 10° 30’ while that of the top has an
elevation of 14° 15’. If the observation point is 250 m from the tree, what is the height
of the tree ?
8. From a pilot’s cockpit, one edge of a runway
has an angle of depression 19°
45’ while the opposite edge has a depression angle of 16° 15’. If the pilot is 2 323.5 m above
the ground, how long is the runway?
9. Two guy wires are attached to a pole 12 m
above the ground level. They make angles of 43° and
72° with the ground at points which are in a line with the base of the
pole. How long are the wires?
10.
Determine the height of the World Trade Center in New York City if from a
window in a
building 91.5 m away the angle of
depression to the base of the Trade Center is 68° while
the angle of elevation to the top is 62°.
11. A light house casts a shadow of 9m when the
angle of elevation of the sun is 56°. How high is the light house?
12. The angle of elevation to a balloon from a
point on the ground is 32°. After a
vertical ascent of
68 m the angle of depression from the balloon to the same point on the
ground is 47°. What are the original and present
heights of the balloon?
13.
From a point 25 m from the base of a tower, a bird flies to its top in a
straight path at an
angle of elevation of 75° 32’. How long is the flight path of the bird ?
What is the height of
the tower ?
14. From an airplane flying at 2,350 m above the
ground, one edge of a runway has an angle of depression 19.5° while the opposite edge has a
depression angle of 15.25°.
How long is the runway?
TRIGONOMETRIC
EQUATIONS
Example.
1. Find the solution of the equation 2
sin x – 1 = 0.
2 sin x – 1 = 0
2 sin x = 1
sin x = ½
x = 30°
or 150°
2. 2 sin x + 1 = 0
2 sin x = – 1
sin x = – ½
x = 330°
or 210°
3. tan x
+ 1 = 0
4. tan x
= 1
When
equations are factorable, factor the equation, equate factors to zero
and solve for the value of the angle. Then check for all allowable values of
the angle.
5. 2 cos2 x – cos x = 1
2 cos2 x –
cos x – 1 = 0
( 2 cos x + 1 ) ( cos
x – 1 ) = 0
2 cos x + 1 = 0 cos x – 1 = 0
2 cos x =
– 1 cos x
= 1
cos x = –
½ x =
0° or
360°
x =
120° or 240°
6. tan2 x – 3 = 0
7. sin2 x – 1 = 0
8. 4 sin 2 x – 3 =
0
9. sin2 x = sin
x
10. cos2 x = ¼
11. 3 tan2 x – 1 = 0
12. 2
cos2 x – cos x
= 0
End of Lesson 1
No comments:
Post a Comment