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.
P = a + b + c
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
RIGHT TRIANGLES
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
Examples : Find the missing side, the perimeter and the
area of the given right triangle.
1. a = 12
cm , c =
13 cm
2. b = 12 cm ,
a = 9 cm
3. c = 35 cm, b = 21
cm
4. a = 36 cm,
b = 48 cm
5. a = 81 cm,
c = 135 cm
6. c = 12 cm, a
= 6 cm
7.
Will a round glass table top, 2.5 m in diameter, fit through a doorway
which is 2.13 m high and 0.96 m
wide ?
8.
A rectangle has a base of 12 cm. If the diagonal is 15 cm, determine
the
altitude of the rectangle.
9.
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°.
Examples :
Find a
or b.
1. a = 36° 24’ 32’’ 5. b = 56°
43’ 27’’
2. b =
68° 13’ 6. a = 62° 54’ 12’’
3. a =
47° 19’
7. b = 73° 24’ 32’’
4. a =
17° 21’ 36” 8. a = 16° 48’
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.
Ratios
AX = BY = CZ = opposite side =
sine a = sin a .
OX OY OZ hypotenuse
OA = OB = OC = adjacent
side =
cosine a = cos a .
OX OY OZ hypotenuse
AX = BY = CZ = opposite side = tangent
a = tan a .
OA OB OC adjacent side
OA = OB = OC = adjacent side = cotangent
a = cot a .
AX BY CZ opposite side
OX = OY = OZ = hypotenuse
= secant
a = sec a
.
OA OB OC adjacent side
OX = OY = OZ = hypotenuse = cosecant
a
= csc a .
AX BY CZ opposite side
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 q = 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 q = 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 60°. At high tide the angle of elevation to the
top of the tree is 30°.
How high does the water level rises during the high tide along the
line
perpendicular to the shore ?
14.
Determine the height of the World Trade Center in New York City if from a
window
in a
is 68° while the angle of elevation to the top is 62°.
15. A light house
casts a shadow of 9m when the angle of elevation of the sun is 56°.
How high
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 ?
