Lesson 4.2: Trigonometry Ratios and Special Angles
Objectives:
Define angles in standard position and the trigonometric ratios of these
angles
State the CAST Rule
Define positive and negative angles
Define principal angle, related acute angle, and co-terminal angles
State Trigonometric Ratios of Special Angles (300- 450 – 600)
Solving Problems
I. ANGLES IN A CIRCLE:
A. ANGLES IN STANDARD POSITION:
An angle is in STANDARD POSITION when it is centred at the origin, the initial arm
is the positive x-axis and the terminal arm rests anywhere within the four quadrants
Θ > 0 (positive) when the terminal
arm rotates counter-clockwise
Θ < 0 (negative) when the terminal
arm rotates clockwise.
QII
QI
Terminal arm
r2 = x2 + y2
Initial arm
P(x, y)
QIII
QIV
Sin Θ = y / r
Cos Θ = x / r
Tan Θ = y / x
B. PRINCIPAL ANGLE: The angles between O0 and 3600
C. RELATED ACUTE ANGLE: The angles between O0 and 900, and formed by the terminal arm
and the x-y axis.
D. CO-TERMINAL ANGLES: These angles have the same initial arms and terminal arms, but
having different revolutions.
QII (S)
Terminal arm
QI (A)
Principal Angle = 210º
Initial arm
Related Acute Angle = 30º
QIV
(C)
QIII
(T)
COTERMINAL ANGLES = (ANGLE IN STANDARD POSITION)(k.2  )
K: integers (…-3, -2, -1, 0, 1, 2, 3,…)
Examples:
Positive Co-terminal Angle
210º + 360º = 570º
570º + 360º = 930º
Negative Co-terminal Angle
210º - 360º = -150º
-150º - 360º= -510º
CAST Rule:
S A
Sine is positive
All others are negative
All are positive
TC
Tangent is positive
All others are negative
Cosine is positive
All others are negative
II. SPECIAL ANGLES (300 - 450 - 600):
600
2
sin 300 = O/H = 1/ 2
cos 300 = A/H = V3 / 2
tan 300 = O/A = 1 / V3
1
300
900
V3
450
V2
450
Angle
SINE
COSINE
TANGENT
Examples:
a) sin 3П/4
1
1
sin 600 = V3 / 2
cos 600 = 1 / 2
tan 600 = V3
sin 450 = cos 450 = 1
V2
tan 450 = 1
900
300 (П/6)
1/ 2
V3/ 2
1/ V3
Find the exact value of
450 (П/4)
V2/ 2
V2/ 2
1
600 (П/3)
V3/ 2
1/ 2
V3
b) tan 5П/6
c) sec 4П/3
d) csc 3П/4
SOLUTION:
a) Since the related acute angle of 3П/4 is П/4, therefore,
the point P(x, y) on the terminal arm will be P(-1, 1) and r = V2.
We have: sin 3П/4 = y/ r = 1/ V2
(QII)
b) Since the related acute angle of tan 5П/6 is П/6, therefore
the point P(x, y) on the terminal arm will be P(-V3, 1) and r= 2
We have: tan 5П/6 = y/ x = -1/ V3
(QII)
c) Since the related acute angle of sec 4П/3 is П/6, therefore
the point P(x, y) on the terminal arm will be P(-1, -V3) and r= 2
We have: sec 4П/ 3 = 1/ cos 4П/ 3 = 1/ x/ r = r/ x = 2/ -1 = -2
d) csc 3П/4 = 1/ sin 3П/ 4 = 1/ 1/ V2 = V2
(QII)
Practice 1: Special Angles
Name ____________________
Date _____________________
Knowledge
For each function, find the quadrant containing the angle, the related acute angle, and
the exact value of the given function:
ANGLE
5
4
7
2. sec
4
5
3. tan
6
1.
Quadrant
Related Acute
Value
sin
Application/Communication
4. a. Find the angle θ created by the intersection of the unit circle and radius with
point P, as shown below.
b. What are the coordinates of point P where the line y = ½ intersects the unit
circle?
c. Find the angle created by the intersection of the unit circle and radius with
point Q, as shown below.
d. What are the coordinates of point Q where the line y = ½ intersects the unit
circle?
e. Explain how this shows that if sinθ = ½, cosθ =
Q
P
θ

3
2
y 
1
2
Radians and Special Angles (Answers)
ANGLE
5
4
7
2. sec
4
5
3. tan
6
1.
sin
4. a.
Quadrant
III
I
II
Related Acute

4

4

6
Value
1
2
1
2
1

3

b.
1/2
1
sin   2
1
1
sin  
2
  30
cos
cos

6
6

x
1
3
x
2
 3 1
P
 2 , 2 


c. Related acute angle
d.


1

5
, yielding principal angle of
.
6
6
x
1
1
1/2
3
x
2
 
6
In quadrant II, the value of x is negative.
x
 3 1
 2 , 2 


Q
e. Sine is positive in quadrants I & II. Cosine is positive in quadrant I, and negative in
quadrant II. Using the related acute angle of
value of ½, and a cosine value of
3
.
2

in both quadrants I & II yields a sine
6
Practice 2: Radians and Special Angles
Name ____________________
Date _____________________
Knowledge
For each function, find the quadrant containing the angle, the related acute angle, and
the exact value of the given function:
ANGLE
5
4
7
2. sec
4
5
3. tan
6
1.
Quadrant
Related Acute
Value
sin
Application/Communication
5. a. Find the angle θ created by the intersection of the unit circle and radius with
point P, as shown below.
b. What are the coordinates of point P where the line y = ½ intersects the unit
circle?
c. Find the angle created by the intersection of the unit circle and radius with
point Q, as shown below.
d. What are the coordinates of point Q where the line y = ½ intersects the unit
circle?
e. Explain how this shows that if sinθ = ½, cosθ =
Q
P
θ

y 
Radians and Special Angles (Answers)
3
2
1
2
ANGLE
Quadrant
III
5
4
7
2. sec
4
5
3. tan
6
1.
sin
II
b.
1
sin   2
1
1
sin  
2
  30
cos
cos

6

6


1
2
1
2
1

3

x
1
3
x
2
 3 1
P
 2 , 2 


c. Related acute angle
Value

4

4

6
I
4. a.
d.
Related Acute
1
1/2
x

5
, yielding principal angle of
.
6
6
x
1
1
1/2
3
x
2
 
6
In quadrant II, the value of x is negative.
x
 3 1
 2 , 2 


Q
e. Sine is positive in quadrants I & II. Cosine is positive in quadrant I, and negative in
quadrant II. Using the related acute angle of
value of ½, and a cosine value of
3
.
2

in both quadrants I & II yields a sine
6