Download Trigonometric Ratios for any Angle

T3
MATHEMATICS
SUPPORT CENTRE
Title:
Trigonometric Ratios for any Angle
Target: On completion of this worksheet you should know the definitions for the
sine, cosine and tangent of any angle and be able to write these in terms of an acute
angle.
Consider a circle of radius r :
y
Suppose OP has turned through an angle
of 2300
P (x,y)
y
x
O
N
N
O
x
P (x,y)
Suppose OP is a rotating arm starting
along the x-axis and turning anti-clockwise:
OP = r
(radius of circle)
PN = y
(y co-ordinate of point P)
ON = x
(x co-ordinate of point P)
Let the angle turned through be θ 0
ie ∠ PON = θ 0
We define:
PN y
=
OP r
ON x
cosθ =
=
OP r
PN y
tanθ =
=
ON x
sin θ =
These definitions are consistent with our
previous definitions for right angled
triangles.
Note: See graph sheet G14 for the graphs
of sine, cosine and tangent.
Mathematics Support Centre,Coventry University, 2000
The point P has both the x and y coordinates negative so using the definitions
we have:
y
<0
( r > 0)
r
x
cos 230 0 = < 0
r
y
tan 230 0 = > 0
x
In fact sin 2300 = -0·766. Use your
calculator to check that cos 2300 is
negative and tan 2300 is positive.
Also considering triangle OPN,
∠NOP = 2300 – 1800 = 500
y
PN
sin 500 =
=
OP r
so sin 2300 = - sin 500 (check this result)
Similarly cos 2300 = - cos 500 and
tan 2300 = tan 500
sin 230 0 =
If the angle turned through by OP is θ then
if θ is between:
00 and 900 it is in the first quadrant
900 and 1800 it is in the second quadrant
1800 and 2700 it is in the third quadrant
2700 and 3600 it is in the fourth quadrant
0
rd
So 230 is in the 3 quadrant and for any
angle in this quadrant the sine and cosine
are negative and the tangent is positive.
Similarly we find the following:
sin θ
+
+
-
Quadrant
1st
2nd
3rd
4th
cos θ
+
+
tan θ
+
+
-
This can be shown in a diagram as follows:
Sin
only positive
All positive
Tan
only positive
Cos only positive
Exercise
Without using a calculator state which of
the following are positive and which are
negative:
1. sin 1500
2. cos 3050
3. tan 950
4. sin 2500
(Answers: positive, positive, negative,
negative)
Examples
Write the following in terms of an acute
angle:
1. cos 1200
1200
This angle is 600 and
since 1200 is in the 2nd quadrant cos 1200
is negative so cos 1200 = - cos 600
2. tan 1900
1900
angle = 1900 - 1800
= 100
1900 is in the 3rd quadrant where tan is
positive so tan 1900 = tan 100
This is usually abbreviated as follows:
S
A
T
C
Examples
Without using a calculator state which of
the following are positive and which are
negative:
0
1. cos 150
This is in the 2nd quadrant where cos is
negative so cos 1500 < 0
2. tan 2100
This is in the 3rd quadrant where tan is
positive so tan 2100 > 0
3. sin 4000
This is in the 1st quadrant (one complete
turn and then 400) so sin 4000 > 0
Mathematics Support Centre,Coventry University, 2000
To find a trig ratio in terms of an acute
angle:
• Mark the angle on a sketch
• Find the acute angle made with the
horizontal ie the 00/1800/3600 line
• Use the diagram opposite to find
whether the answer is positive or
negative
Exercise
Write the following in terms of an acute
angle:
1. sin 3000
2. sin 2340
0
3. tan 100
4. cos 1450
5. cos 3900
6. sin 950
0
7. tan 285
8. tan 2500
9. cos 3100
10. sin 4200
0
11. tan 740
12. cos 2750
(Answers: -sin 600, -sin 540, -tan 800,
-cos 350, cos 300, sin 850, -tan 750,
tan 700, cos 500, sin 600, tan 200, cos 850)