IN FOUR QUADRANTS
• The four quadrants
• Co-ordinates in the first quadrants
• Trigonometry ratios in the first
quadrants
Y axis
90°
180°
second
quarter
In trigonometry we
work with the right
angled triangles and
the four quarters
First quarter
𝑂
third quarter
270°
Fourth
quarter
0° X axis
360°
We call the quarters
quadrants
Y axis
One
full
revolutions
is 360°
180°
90°
𝑂
270°
One quarter revolutions
is 90°
4 × 90° = 360°
0° X axis
360°
Y axis
A(𝑥, 𝑦)
𝑦
𝜃
𝑥
X axis
In the triangle ABC,
AB = 5 units
BC = 3 units
AC = 4 units
How would you prove
that triangle ABC is right
angled?
B
5
A
4
3
C
B
Pythagoras’s theorem states:
The square on the hypotenuse
equals the sum of the squares on
the other two sides
5
A
4
3
C
Pythagoras’s theorem states:
B
The square on the hypotenuse equals the sum of
the squares on the other two sides
5
In this triangle:
Is 𝐴𝐵 = 𝐴𝐶 + 𝐵𝐶 ?
2 =
2
2
5
4 + 3
25 = 16 + 9
25 = 25 TRUE
2
2
2
A
4
3
C
This means ∆ ABC is
right angled.
Draw the basic 3, 4, 5 triangle
Draw a circle around it
5
4
3
The radius is the length of the
hypotenuse
The center of the circle is at the letter O
Point O is
called the
origin
Y axis
Draw axes through the origin
(4, 3) What are the co-ordinates of
this point?
5
O
4
3
X axis
Move 4 along X axis
Move 3 along y axis
Co-ordinates of the point are: (4, 3)
sin 𝜃
cos 𝜃
tan 𝜃
sine 𝜃
cosine 𝜃
tangent 𝜃
In the diagram, can you work out the trig ratios for angle 𝜃 in terms of
the sides of the triangle?
Y axis
We need to label the opposite
( 𝑥, 𝑦) adjacent sides, and the hypotenuse
𝑟H
𝑦 O
𝜃 𝑥
X axis
O
A
Co-ordinates of the point are:(𝑥, 𝑦)
y
O
tan 𝜃 =
=
A
x
y
O
sin 𝜃 =
H
=
r
x
A
cos 𝜃 =
=
H
r
What happens to x and y as angle 𝜃 decreases?
90°
A(𝑥, 𝑦 )
𝑟
O
𝑦
𝜃
𝑥
Watch what happens
when 𝜃 changes.
0°
When 𝜃 = 0°, how big is
side y?
y=r
y=0
y=x
Example 1 – Evaluating Trigonometric Functions
• Let (–3, 4) be a point on the terminal side of 
Find the sine, cosine, and tangent of .
by using the Pythagorean Theorem and
the given point that x = –3, y = 4, and
2
+ 42
𝑟=
−3
𝑟=
9 + 16
𝑟 = 25
𝑟= 5
x = –3, y = 4, and r = 5
y
O
tan 𝜃 =
=
A
x
y
O
sin 𝜃 =
H
=
r
x
A
cos 𝜃 =
=
H
r
4
4
=
=−
3
–3
4
4
=
=
5
5
=
–3
5
3
=−
5
If (4, -8) is a point on the terminal side of
angle 𝛼 in standard position, evaluate the six
trigonometric functios of 𝛼.
by using the Pythagorean Theorem and
the given point that x = 4, y = -8, and
2
4
+ −8
𝑟=
𝑟 = 16 + 64
𝑟 = 80
𝑟=4 5
2
x = 4, y = -8, and r = 4 5
y
O
tan 𝜃 =
=
A
x
y
O
sin 𝜃 =
H
=
r
x
A
cos 𝜃 =
=
H
r
=
=
=
-8
4
-8
4 5
4
4 5
= −2
=
=
−2
5
1
5
×
×
5
2 5
=−
5
5
5
5
=
5
5
x = 4, y = -8, and r = 4 5
x
A
cotan 𝜃 =
=
O
y
r
H
csc 𝜃 =
O
=
y
r
H
sec 𝜃 =
=
A
x
=
4
-8
1
=−
2
5
4 5
=
=− 2
−8
4 5
=
= 5
4
Example 2
Evaluate trigonometric functions
8
−
17
If cos 𝜃 =
𝑎𝑛𝑑
sin 𝜃 < 0 , 𝑓𝑖𝑛𝑑 tan 𝜃 𝑎𝑛𝑑 csc 𝜃
Solutions
8
−
17
If cos 𝜃 =
𝑎𝑛𝑑
sin 𝜃 < 0 , 𝑓𝑖𝑛𝑑 tan 𝜃 𝑎𝑛𝑑 csc 𝜃
Cos –
Sin H
quadrant III
x
A
cos 𝜃 =
=
H
r
r
H
csc 𝜃 =
=
O
y
=
-8
=
−15
O
𝜃
17
17
17
8 A
17
= − 15
𝑦=
𝑟2 − 𝑥2
𝑦 = 172 − 82
𝑦 = 225
𝑦 = 15 (−)
If sin 𝜃 = −
cos 𝜃.
5
3
and cos 𝜃 > 0, find tan 𝜃 and
https://www.andrews.edu/~rwright/Precalculus-RLW/Text/04-05.html