What has to be broken before it can be
used?
5.3 and 5.4 Understanding
Angles Greater than 90
y
Terminal Arm
θ
Initial Arm
x
We're going to explore how triangles in a
Cartesian plane have trig ratios that relate to
each other
Angles, Angles, Angles
• An angle is formed when a ray is rotated about a
fixed point called the vertex
Terminal Arm (the part
that is rotated)
Vertex
θ
Initial Arm (does
not move)
Think of Hollywood
Depending on
how hard the
director wants to
snap the device,
he/she will vary
the angle
between the
initial arm and
the terminal arm
Terminal Arm
Initial Arm
• The trigonometric ratios have been
defined in terms of sides and acute
angles of right triangles.
• Trigonometric ratios can also be defined
for angles in standard position on a
coordinate grid.
y
x
Coordinate grid
Standard Position
• An angle is in standard position if the vertex of
the angle is at the origin and the initial arm lies
along the positive x-axis. The terminal arm can
be anywhere on the arc of rotation
y
Terminal Arm
θ
Initial Arm
x
Greek Letters such
as α,β,γ,δ,θ (alpha,
beta, gamma, delta,
theta) are often used
to define angles!
For example……
Terminal Arm
Terminal Arm
Initial arm
Initial arm
Not Standard Form
Terminal Arm
Initial arm
Standard Form
Positive and Negative angles
Positive angles
Negative angles
θ
θ
A positive angle is
formed by a
counterclockwise
rotation of the
terminal arm
A negative angle is formed
by a clockwise rotation of
the terminal arm
The Four Quadrants
The x-y plane is divided
into four quadrants. If
angle θ is a positive angle,
then the terminal arm lies
in which quadrant?
Quadrant II
Quadrant I
Quadrant III
Quadrant IV
0º< θ < 90º
90º < θ < 180º
180º < θ < 270º
270 º < θ < 360º
Principal Angle and Related
Acute Angle
The principal angle is the
angle between the initial arm
and the terminal arm of an
angle in standard position. Its
angle is between 0º and 360º
The related acute angle is the acute
angle between the terminal arm of an
angle in standard position (when in
quadrants 2, 3, or 4). and the x-axis.
The related acute angle is always
positive and is between 0º and 90º
Terminal Arm
θ
β
Related Acute
Angle
Principal Angle
Initial Arm
Let’s look at a few examples……
In these examples, θ represents the principal
angle and β represents the related acute angle
θ
β
Principal angle: 65º
θ
Principal angle: 140 º
No related acute
angle because the
principal angle is in
quadrant 1
Related acute angle: 40º
θ
θ
β
Principal angle: 225º
Related acute angle: 45º
β
Principal angle: 320º
Related acute angle: 40º
Notice anything?
•In the first quadrant the principal angle and
related acute angle are always the same
•In the second quadrant we get the principal angle
by taking (180º - related acute angle)
•In the third quadrant we can get the principal
angle by taking (180º + related acute angle)
•In the fourth quadrant we can get the principal
angle by taking (360º - related acute angle)
Let’s work with some numbers!
Angles
Quadrant
Sine
Ratio
Cosine
Ratio
Tangent
Ratio
Principal angle
60º
Related acute
angle (none)
Principal angle
135º
Related acute
1
0.8660
0.5
1.7320
2
angle 45º
Principal angle
220º
Related acute
angle 40º
Principal angle
300º
Related acute
angle 60º
3
4
0.7071
-0.7071
-1
0.7071
0.7071
1
-0.6427
-0.7760
0.8391
0.6427
0.7760
0.8391
-0.8660
0.5
-1.7320
0.8660
0.5
1.7320
Quadrant 1
Sinθ is positive
Cosθ is positive
Tanθ is positive
θ
Quadrant 2
Sinθ = sin (180° - θ)
-Cosθ = cos (180° - θ)
-Tanθ = tan (180° - θ)
(180° - θ)
θ
Quadrant 3
-Sinθ = sin (180° + θ)
-Cosθ = cos (180° + θ)
Tanθ = tan (180° + θ)
(180° + θ)
θ
Quadrant 4
-Sinθ = sin (360° - θ)
Cosθ = cos (360° - θ)
-Tanθ = tan (360° - θ)
(360° - θ)
θ
Summary
Only sine is positive
All ratios are positive
S
A
Only tangent is positive
Only cosine is positive
T
C
For any principal angle greater than 90, the values of the
primary trig ratios are either the same as, or the negatives of,
the ratios for the related acute angle
When solving for angles greater than 90, the related acute
angle is used to find the related trigonometric ratio. The CAST
rule is used to determine the sign of the ratio
CAST Rule
Quadrant II
Sine
1800 - q
1800 + q
Tangent
Quadrant III
Quadrant I
All
q
3600 - q
Cosine
Quadrant IV
Example1.
Point P(-3,4) is on the terminal arm of an angle in standard
position.
a)Sketch the principal angle θ
b) Determine the value of the related acute angle to the
nearest degree
c) What is the measure of θ to the nearest degree?
Solution
a) Point P(-3,4) is in quadrant 2, so the principal
angle θ terminates in quadrant 2.
P(-3,4)
4
θ
β
-3
b) The related acute angle β can be used as part of a
right triangle with sides of 3 and 4. We can figure
out β using SOHCAHTOA.
Sin 
opp
hyp
4

5
q  180 - 
 180 - 53
 127
4
5
  53
  sin -1  
Note…..Whenever we make a triangle such as the one
above there is something important to remember…
THE HYPOTENUSE will always be expressed as a
positive value, regardless of the quadrant in which it
occurs!! Lets look at an example….
Example 2
Point (3,-4) is on the terminal arm of an angle in standard position
a) What are the values of the primary trigonometric functions?
b) What is the measure of the principal angle θ to the nearest
degree?
Assuming that you can draw a
circle around the x-y axis, with
your point lying somewhere on
the perimeter, then it would
follow that the hypotenuse of our
right angled triangle would be the
same as the radius of the circle.
Solution
θ
Using pythagorean theorem, we find that r = 5
(note it is positive regardless of the quadrant.
Using these values, then
3
sin B 
-4
r =5
-
4
5
y
r
cos B 

3
5
x
r
t an B 
-
y
x
4
3
To evaluate B, select cosine and solve for B. Using cos B gives us
 3
B  cos-1  
5

B  53
……….
From the sketch, clearly θ is not 53°. This angle is the
related acute angle. In this case θ = 360°-53° = 307°
Just as a side note….once again notice that if you take the
cos of 307° you get 0.6018 and if you take the cos of 53°
you also get 0.6018
Ok, hmk
Well Ross,
what is
it?
Wait for it,
wait for it…
WOOOOW!