TRIGONOMETRY OF
RIGHT TRIANGLES
TRIGONOMETRIC RATIOS
Consider a right triangle with  as one of its acute angles. The
trigonometric ratios are defined as follows .
hypotenuse

opposite
opposite
sin  =
hypotenuse
hypotenuse
csc  =
opposite
adjacent
cos  =
hypotenuse
hypotenuse
sec  =
adjacent
opposite
tan  =
adjacent
adjacent
cot  =
opposite
adjacent
Note: The symbols we used for these ratios are abbreviations
for their full names: sine, cosine, tangent, cosecant, secant
and cotangent.
RECIPROCAL FUNCTIONS
The following gives the reciprocal relation of the six
trigonometric functions.
sin  =
1
csc
cos  =
1
sec
tan  =
1
cot 
csc  =
1
sin
sec  =
1
cos 
cot  =
1
tan
THE PYTHAGOREAN THEOREM
The Pythagorean Theorem states that the square of the
hypotenuse is equal to the sum of the squares of the other
two sides. In symbol, using the ABC as shown,
c  a b
2
B
a
C
2
2
c
b
A
EXAMPLE:
1. Draw the right triangle whose sides have the
following values, and find the six trigonometric
functions of the acute angle A:
a) a=5 , b=12 , c=13
EXAMPLE:
1. Draw the right triangle whose sides have the
following values, and find the six trigonometric
functions of the acute angle A:
b) a=1 , b= 3 , c=2
EXAMPLE:
2. The point (7, 12) is the endpoint of the terminal
side of an angle in standard position. Determine
the exact value of the six trigonometric functions
of the angle.
EXAMPLE:
3. Find the other five functions of the acute angle A,
given that:
3
a) tan A =
4
EXAMPLE:
3. Find the other five functions of the acute angle A,
given that:
b) sec A = 2
EXAMPLE:
3. Find the other five functions of the acute angle A,
given that:
2mn
c) sin A = 2 2
m n
FUNCTIONS OF COMPLIMENTARY ANGLES
a
sin A =
c
a
cos B =
c
b
cos A =
c
b
sin B =
c
a
tan A =
b
a
cot B =
b
b
cot A =
a
b
tan B =
a
b
sec A =
c
b
csc B =
c
c
csc A =
a
c
sec B =
a
B
a
C
c
b
A
Comparing these formulas for
the acute angles A and B, and
making use of the fact that A
and B are complementary
angles (A+B=900), then
FUNCTIONS OF COMPLIMENTARY ANGLES
sin B = sin (900  A) = cos A
cos B = cos (900  A) = sin A
tan B = tan (900  A) = cot A
cot B = cot (900  A) = tan A
0
(
90
 A) = csc A
sec B = sec
0
(
90
 A) = sec A
csc B = csc
The relations may then be expressed by a single
statement: Any function of the complement of an
angle is equal to the co-function of the angle.
EXAMPLE:
4. Express each of the following in terms of its
cofunction:
0
0
'
"
0
a) sin76 b) csc 80 35 32 c) tan(A  15 )
EXAMPLE:
5. Determine the value of  that will satisfy the ff.:
0
a) csc (6  12 ) = sec 7
1
0
b) sin (4   5 ) =
0
sec(3  10 )
TRIGONOMETRIC FUNCTIONS OF SPECIAL
ANGLES 450, 300 AND 600
To find the functions of 450, construct a diagonal in a
square of side 1. By Pythagorean Theorem this
diagonal has length of 2 .
1
sin
= 
2
1
0
cos 45 = 
2
450
450
2
1
450
1
tan 450 = 1
2
2
2
2
csc 450 = 2
sec 450 = 2
cot 450 = 1
To find the functions of 300 and 600, take an
equilateral triangle of side 2 and draw the bisector of
one of the angles. This bisector divides the
equilateral triangle into two congruent right triangles
whose angles are 300 and 600. By Pythagorean
Theorem the length of the altitude is 3 .
300
2
3
600
1
sin
300
1
=
2
cos
300
3
=
2
tan
300
1
3
= 
3
3
cot 300 = 3
sec
300
2 2 3
= 
3
3
csc 300 = 2
cos
600
1
=
2
sin
600
3
=
2
cot
600
1
3

=
3
3
tan 600 = 3
csc
600
2 2 3
= 
3
3
sec 600 = 2
EXAMPLE:
6. Without the aid of the calculator, evaluate the
following:
a) 3 tan2 600 + 2 sin2 300 – cos2 450
b) 5 cot2 450 + 5 tan 450 + sin 300
c) cos2 600 – csc2 300 – sec 300
d) tan 600 + 2 cot 300 – sin 600
e) tan5 450 + cot2 450 – sin4 600
EXAMPLE:
6. Without the aid of the calculator, evaluate the
following:
a) 3 tan2 600 + 2 sin2 300 – cos2 450
EXAMPLE:
6. Without the aid of the calculator, evaluate the
following:
b) 5 cot2 450 + 5 tan 450 + sin 300
EXAMPLE:
6. Without the aid of the calculator, evaluate the
following:
c) cos2 600 – csc2 300 – sec 300
EXAMPLE:
6. Without the aid of the calculator, evaluate the
following:
d) tan 600 + 2 cot 300 – sin 600
EXAMPLE:
6. Without the aid of the calculator, evaluate the
following:
e) tan5 450 + cot2 450 – sin4 600
Find….
1. sin 32 o
=
2. cos 81 o
=
3. tan 18 o
=
4. sec 58 o
=
5. cot 78 o
=
Use Trigonometry To Find Angles
IF sin  = 0.2588 find 
IF cos  = 0.3746 find 
IF tan  = 4.011 find 
B
c
A
a
c
a
34
C
19 ซม.


…………………..
…………………..
Use trigonometric about special right triangles to find the
value of x and y.
Use trigonometric about special right triangles to find the
value of x and y.
Find the missing lengths
Find the missing lengths
Trigonometric Word Problems
*The angle between the HORIZONTAL and a line of
sight is called an angle of elevation or an angle of
depression
A 20-foot ladder is leaning against a wall. The base of the ladder is 10 feet
from the wall. What angle does the ladder make with the ground
20
?
10
Cos A =
𝟏𝟎
𝟐𝟎
Cos A =
𝟏
𝟐
A = 60°
How tall is a bridge if a 6-foot tall person standing 100 feet away can see
the top of the bridge at an angle of 60 degrees to the horizon?
60°
100
6
A hot air balloon is flying at an altitude of 1500 m. The angle of
depression from the balloon to a landmark on the ground is 30º.
a) What is the balloon’s horizontal distance to the landmark, to
the nearest metre?
b) What is the balloon’s direct distance to the landmark, to the
nearest metre?
Two buildings are 30 m apart. The angle from the top of the shortest
building to the top of the taller building is 30°. The angle from the top of
the shorter building to the base of the taller building is 45°. What is the
height of the taller building to the nearest metre?