Understanding Angles

1
5.3 and 5.4 Evaluating Trig Ratios for Angles
between 0 and 360
What has to be broken before it can be used?
2
5.3 and 5.4 Understanding Angles Greater than
y
Terminal Arm
?
x
Initial Arm

• We're going to explore how triangles in a
  Cartesian plane have trig ratios that relate to
  each other

3
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
?
Initial Arm (does not move)
4
Think of Hollywood
Terminal Arm
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

Initial Arm
5

• 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.

Coordinate grid
6
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
Greek Letters such as a,ß,?,d,? (alpha, beta,
gamma, delta, theta) are often used to define
angles!
Terminal Arm
?
x
Initial Arm
7
For example
Terminal Arm
Initial arm
Not Standard Form
Terminal Arm
Initial arm
8
Positive and Negative angles
Positive angles
Negative angles
?
?
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
9
The Four Quadrants
The x-y plane is divided into four quadrants. If
angle ? is a positive angle, then the terminal
arm lies in which quadrant?
Quadrant I
Quadrant II
0ºlt ? lt 90º
Quadrant III
Quadrant IV
90º lt ? lt 180º
180º lt ? lt 270º
270 º lt ? lt 360º
10
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
?
Principal Angle
ß
Related Acute Angle
Initial Arm
Lets look at a few examples
11
In these examples, ? represents the principal
angle and ß represents the related acute angle
?
?
ß
Principal angle 65º
Principal angle 140 º Related acute angle 40º
No related acute angle because the principal
angle is in quadrant 1
?
?
ß
Principal angle 225º Related acute angle 45º
ß
Principal angle 320º Related acute angle 40º
12
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)

13
Lets work with some numbers!
Angles Quadrant Sine Ratio Cosine Ratio Tangent Ratio








Principal angle 60º
1
0.8660
0.5
1.7320
Related acute angle (none)
Principal angle 135º
2
0.7071
-0.7071
-1
Related acute angle 45º
0.7071
0.7071
1
Principal angle 220º
3
-0.6427
-0.7760
0.8391
Related acute angle 40º
0.6427
0.7760
0.8391
Principal angle 300º
-0.8660
0.5
-1.7320
4
Related acute angle 60º
0.5
0.8660
1.7320
14
Quadrant 1
Sin? is positive
Cos? is positive
Tan? is positive
?
15
Quadrant 2
Sin? sin (180 - ?)
-Cos? cos (180 - ?)
-Tan? tan (180 - ?)
(180 - ?)
?
16
Quadrant 3
-Sin? sin (180 ?)
-Cos? cos (180 ?)
Tan? tan (180 ?)
(180 ?)
?
17
Quadrant 4
-Sin? sin (360 - ?)
Cos? cos (360 - ?)
-Tan? tan (360 - ?)
(360 - ?)
?
18
Summary
All ratios are positive
Only sine is positive
S
A
Only cosine is positive
Only tangent is positive
T
C
19
Summary

• 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 I
Quadrant II
Sine
All
1800 - q
q
3600 - q
1800 q
Cosine
Tangent
Quadrant IV
Quadrant III
20
Example1.
Point P(-3,4) is on the terminal arm of an angle
in standard position. a)Sketch the principal
angle ? b) Determine the value of the related
acute angle to the nearest degree c) What is the
measure of ? to the nearest degree?
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Solution
a) Point P(-3,4) is in quadrant 2, so the
principal angle ? terminates in quadrant 2. b)
The related acute angle ß can be used as part of
a right triangle with sides of 3 and 4. We can
figure out ß using SOHCAHTOA.
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.
22
Example 2

• Point (3,-4) is on the terminal arm of an angle
  in standard position
• What are the values of the primary trigonometric
  functions?
• What is the measure of the principal angle ? to
  the nearest degree?

23
Solution
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.
Using pythagorean theorem, we find that r 5
(note it is positive regardless of the quadrant.
Using these values, then
To evaluate B, select cosine and solve for B.
Using cos B gives us
.
24
From the sketch, clearly ? is not 53. This
angle is the related acute angle. In this case ?
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
25
Ok, hmk is pg. 299 1-6, 8,-10
Wait for it, wait for it
Well Ross, what is it?
WOOOOW!