2. TRIGONOMETRIC
RATIOS
Consider a right triangle with  as one of its acute
angles. The trigonometric ratios are defined as follows .
opposite
sin  =
hypotenuse

adjacent
opposite
adjacent
cos  =
hypotenuse
hypotenuse
sec  =
adjacent
opposite
tan  =
adjacent
hypotenuse
hypotenuse
csc  =
opposite
adjacent
cot  =
opposite
Note: The symbols we used for these ratios are abbreviations
for their full names: sine, cosine, tangent, cosecant, secant
and cotangent.

3. RECIPROCAL
FUNCTIONS
The following gives the reciprocal relation of
the six trigonometric functions.
sin  =
1
csc
cos  =
1
sec
1
tan  =
cot 
csc  =
1
sin
sec  =
1
cos 
1
cot  =
tan

4. 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,
B
a
C
c 2  a2  b2
c
b
A

5. FUNCTIONS OF COMPLIMENTARY
ANGLES
a
a
sin A =
c
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

6. FUNCTIONS OF COMPLIMENTARY
ANGLES
sin B = sin (900  A) = cos
cos B = cos (900  A) = sin A
tan B = tan (900  A) = cot A
cot B = cot (900  A) = tan A
sec B = sec (900  A) = csc A
csc B = csc (900  A) = sec A
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.

7. 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
0=1 
cos 45
2
450
1
450
2
1
450
tan 450 = 1
2
2
2
2
csc 450 = 2
sec 450 = 2
cot 450 = 1

8. 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

9. 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

10. 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
b) a=1 , b= 3 , c=2
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.

11. 3. 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

12. 
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troduction-to-trigonometry