Download Outcome T1 Angles in Standard Position McGraw

T1: Angles in Standard Position
Outcome T1
Angles in Standard Position
McGraw-Hill 2.1
Key Terms:
Initial Arm
Terminal Arm
Angle in Standard Position
Reference Angle
Exact Value
Pre-Calculus Math 30S - Notes
T1: Angles in Standard Position
Pre-Calculus Math 30S - Notes
Lesson 2.0: Trigonometry Review
Triangles are labeled by their ____________: ΔABC
Angles can be named one of two ways:
 by the specific __________, i.e. angle A would be named  A.

by the three vertices that ______________the angle, i.e. angle A can be named
 BAC or ________________.
Sides can be named one of two ways:
 by the ____________________ that define the side, i.e. AC.

by the lower case letter of the vertex ________________ the side, i.e. side AC is labeled side b.
When referring to sides related to an angle, certain words are used:
The Pythagorean Theorem is:
2
2
2
a b  c
a 2  c 2  b2
or
The three trigonometric ratios are:
opposite
sin  
hypotenuse
cos  
adjacent
hypotenuse
tan  
opposite
adjacent
How do I remember these 3 formulas?
Page 2 of 11
T1: Angles in Standard Position
Pre-Calculus Math 30S - Notes
Example: Solve for x in the following triangles. Round to nearest tenth.
a)
b)
c)
d)
e)
f)
Example 2: Calculate the length of side BC.
A
60° 50°
8 cm
B
C
Homework: Handout “Review – Right Angle Trigonometry”
Page 3 of 11
T1: Angles in Standard Position
Pre-Calculus Math 30S - Notes
Lesson 2.1.1: Angles in Standard Position
ANGLES & QUADRANTS
 In geometry, an angle is formed by two rays with a common endpoint or ____________.

In trigonometry, angles are interpreted as a rotation of a ray.

The starting position is called the ___________________ arm.

The final position is called the ____________________ arm.

If the angle of rotation is __________________ clockwise, then the angle is ____________.

If the rotation is clockwise, the angle is _________________.
ANGLES IN STANDARD POSITION:
An angle in a coordinate plane is in _________________ position if:
a) its vertex is at the ________________.
b) its initial arm is the _________________ x-axis

Angles in standard position are always shown on the Cartesian plane. The
x-axis and the y-axis divide the plane into four _________________.
Quadrant I:
0° < θ < 90°
Quadrant II:
90° < θ < 180°
Quadrant III:
180° < θ < 270°
Quadrant IV:
270° < θ < 360°
Example 3: Sketch each angle in standard position. State the quadrant in which the terminal arm lies.
a) 35°
b) 230°
c) 310°
Page 4 of 11
T1: Angles in Standard Position
Pre-Calculus Math 30S - Notes
REFERENCE ANGLES

For each angle in standard position, there is a corresponding ____________ angle (θ
the __________________ or ___________________ angle.

Symbol:

It is formed between the ___________________ arm and the _________________ x-axis.

The reference angle is always _________________ and measures between _____ and _____.
Quadrant I:
R  
Quadrant II:
 R  180  
  180   R
Quadrant III:
 R    180
  180   R
90°) called
Quadrant IV:
 R  360  
  360   R
Example 1: Determine the reference angle θR for each angle θ. Sketch θ in standard position and label
the reference angle θR.
a) 140°
b) 300°
Homework: McGraw-Hill P. 83 #1-7 (Basic)
#9, 16 (Intermediate)
#15, 17, 20 (Advanced)
Page 5 of 11
T1: Angles in Standard Position
Pre-Calculus Math 30S - Notes
Lesson 2.1.2: Exact Values and Special Triangles

Trig ratios can be used to determine exact values for the families of special angles

These are values that need to be memorized (no _______________)

To determine these exact values, we will use The _________ Circle, where the value of the
radius is always ______. (See next lesson)

Families of angles are those who have the same ____________________ angle.

Any member in the same family will have the same ___________ Trig ratios (ignoring the sign).
45° FAMILY →This family contains all angles that have a reference angle of 45°.
Ex: _______________________________, etc.
c 2  a 2  b2
sin 45 
cos 45 
45°
135°
tan 45 
315°
C1=
1y
Quadrant
sin 
cos 
45°
x1
tan
Page 6 of 11
T1: Angles in Standard Position
Pre-Calculus Math 30S - Notes
60° FAMILY →This family contains angles that have a reference angle of 60°.
Ex: _______________________________, etc.
a  c 2  b2
sin 60 
cos 60 
tan 60 
30°
12
60°
120°
240°
Aa=
300°
Quadrant
sin 
60°
1
1/2
cos 
tan
30° FAMILY →This family contains angles that have a reference angle of 30°.
Ex: _______________________________, etc.
sin 30 
cos 30 
30°
150°
tan 30 
210°
330°
Quadrant
sin 
cos 
tan
60o
1
1
2
30o
3
2
Page 7 of 11
T1: Angles in Standard Position
Pre-Calculus Math 30S - Notes
When dealing with these special triangles, we are easily able to determine the ____________ values of
the trigonometric functions. (No calculator needed!!)
Memorize!!!
Chart:
Special Triangles
sin 
cos 
tan
30°
30°
OR
2
1=
A
3
60°
/
2
45°
60°
1
Example: Find the exact value of the following:
2
C=
1
45°
1
sin 225 
1) Determine which quadrant the angle lies and
determine the related angle.
2) What is the trigonometric ratio of that related
angle (from memory)?
What sign will it have?
Example: Find the exact value of the following trigonometric function:
a) sin 120 
b) cos 210 
c) tan 330 
d) sin 315 
e) cos 45 
f)
tan 240 
Page 8 of 11
T1: Angles in Standard Position
Pre-Calculus Math 30S - Notes
Lesson 2.1.3: The Unit Circle
The __________________________ is a circle with radius of 1 and centered at the origin.
Any point on the Unit Circle can be written
radius is one unit, then:
x=
Therefore
and can be found be using trig ratios! Since the
y=
.
Example: Determine the coordinates of a terminal point on the unit circle for the following angle in
standard position.
Θ = 2400
Page 9 of 11
T1: Angles in Standard Position
Pre-Calculus Math 30S - Notes
CAST Rule:
Page 10 of 11
T1: Angles in Standard Position
Pre-Calculus Math 30S - Notes
90° FAMILY →This family contains all
Ex: _______________________________, etc.
Example: For the following angles in standard position θ, give the exact value requested:
(Without the use of a calculator)
a) sin 270 
b) tan 90 
c) cos180 
d) tan 270 
Homework: McGraw-Hill P. 83 #8 (Basic)
#10 - 13 (Intermediate)
#22, 23 (Advanced)
Page 11 of 11