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Name __________________________________________________ Date ____________________ Block _________
LIP – Phase III – The Unit Circle – 4/27/2020 to 5/1/2020
Remember that we have studied 2 types of special right triangles--
• For the 30 − 60 − 90 triangle:
o The measure of the leg across from the 30° angle is “side”
o The measure of the leg across from the 60° angle is “side ⦁ √3 “
o The measure of the leg across from the right angle is “2 ⦁ side”
• For the 45 − 45 − 90 triangle:
o The measure of the legs across from the 45° angles (there are two of
them) is “side”
o The measure of the leg across from the right angle is “side ⦁ √2 “
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1
2
√3
2
1
2
√3
2
You Try:
You Try:
REVIEW OF SPECIAL RIGHT TRIANGLES
The UNIT CIRCLE is a circle with the center at the origin and a radius of 1 unit.
The unit circle is an easy way to find the exact measure of trig functions for common angles.
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Since special right triangles contain common angles, let the hypotenuse of each
triangle below equal 𝟏 and use the rules of special right triangles (given above) to
find the remaining sides and the trig ratios underneath.
30-60-90 45-45-90
sin 30°: _______ sin 60°: _______ sin 45°: _______
cos 30°: _______ cos 60°: _______ cos 45°: _______
tan 30°: _______ tan 60°: _______ tan 45°: _______
1
30°
1
2
√3
2
1 ?
?
Quad II: (−, +) Quad I: (+, +)
Quad III: (−, −) Quad IV: (+, −)
𝜋
6
𝜋
3
𝜋
4
TRIG FUNCTIONS ON THE UNIT CIRCLE:
sin 𝜃: The exact value is the 𝑦-coordinate of the ordered pair.
cos 𝜃: The exact value is the 𝑥-coordinate of the ordered pair.
tan 𝜃: The exact value is 𝑦 divided by 𝑥.
The signs of (𝒙, 𝒚) in Quadrants I – IV FOR RADIANS:
All multiples of _______ have the same trig ratios as 30°
All multiples of _______ have the same trig ratios as 60°
All multiples of _______ have the same trig ratios as 45°
THE UNIT CIRCLE
• Place the DEGREE angle measure of each angle in the DASHED BLANKS inside the circle.
• Place the RADIAN measure of each angle in the SOLID BLANKS inside the circle.
• Place the COORDINATES of each point in the ordered pairs outside the circle.
*** Quadrants I & II are done for you. Complete Quadrants III & IV. ***
0°
30°
45°
60°
90°
360°
120°
135°
150°
180°
𝜋6⁄
𝜋4⁄
𝜋3⁄
𝜋2⁄
0𝜋
2𝜋3⁄
3𝜋4⁄
5𝜋6⁄
𝜋 1 0
√3
2
1
2
√2
2
√2
2
1
2
√3
2
0 1
−1
2
√3
2
−√3
2
1
2
−√2
2
√2
2
−1 0
2𝜋
90°
270°
360°
(√2
2,√2
2)
√2
2
Remember that
cosine is the 𝑥-
value of the ordered
pair for the angle!
360°
90°
270°
(0, −1)
Remember that
tangent is 𝑦/𝑥 using
the values from the
ordered pair!
−1
0= undefined
(−1
2,√3
2)
−1
2
To determine where to
draw the angle, subtract!
480° − 360° = 120°
EXAMPLE 1:
Sketch the angle, then evaluate the trig function exactly. The first and last problems are done for you.
1. cos 45° _______ 2. sin 30° _______
3. tan 60° ______ 4. sin11𝜋
4 _______ 5. tan 270° _______
EXAMPLE 3:
The cycles of the sine and cosine functions repeat for every multiple of 360° or 2𝜋 radians .
Sketch the angle, then evaluate the trig function exactly. If undefined, write undefined. The first one is done for you.
1. cos 480° _______ 2. sin11𝜋
4 _______ 3. tan
19𝜋
6 ______
45° 0° 180°
270°
0° 180°
480°
Name _______________________________________________ Date ________________________ Block ________
UNIT CIRCLE ASSIGNMENT
Sketch the angle and then evaluate the trig function. Show work where necessary.
1. sin 120° _______ 2. tan 330° _______
3. sin 180° _______ 4. cos4𝜋
3 _______
5. sin7𝜋
6 _______ 6. tan (2𝜋) _______
7. tan3𝜋
2 _______ 8. cos
7𝜋
4 _______
9. cos (−45°) _______ 10. sin8𝜋
3 _______
11. tan (−3𝜋
2) _______ 12. sin
11𝜋
6 _______
(This is on
each side of
the 𝑥- and
𝑦-axes)
𝜋2⁄
(1, 0)
(0, 1)
0𝜋
2𝜋 𝜋
3𝜋2⁄
(−1, 0)
(0, −1)
2𝜋
𝑏
Name __________________________________________________ Date ____________________ Block _________
LIP – Phase III – The Sine Function – 5/4/2020 to 5/8/2020
The significant points of the sine functions are the 𝑦-coordinates from the quadrantals
on the unit circle and the angle of that point on the unit circle. ( angle , 𝑦-coordinate )
Trig graphs are periodic which means that they repeat indefinitely .
STANDARD FORM FOR SINE: 𝑦 = 𝑎 sin 𝑏 (𝑥 − 𝑐) + 𝑑
TERMS:
“𝑎” is the amplitude . This is the distance from the center line to the minimum/maximum.
“𝑏” is used to find the period: period = This is the horizontal length of each cycle.
“𝑐” is the phase shift . This is the horizontal shift/horizontal translation .
“𝑑” is the vertical shift/vertical translation .
amplitude phase shift
period
angle
vertical shift
2𝜋
𝑏=
2𝜋
1= 2𝜋
2𝜋
𝑏=
2𝜋
1= 2𝜋
To find the significant points
for a transformed graph, apply
the transformations to the
standard 5 significant points
from the parent function.
The parent function of sine is 𝑦 = sin 𝑥 .
1. In the parent function, 𝑎 = 1 , 𝑏 = 1 , 𝑐 = 0 , and 𝑑 = 0 . This means that:
a) The amplitude is 1 .
b) The period is ___________
2. The significant points are (0, 0), (𝜋
2 , 1) , (𝜋 , 0), (
3𝜋
2 , −1) , (2𝜋 , 0)
3. The graph of the sine parent function is:
Period: 2𝜋
Domain: [0 , 2𝜋]
Range: [−1 , 1]
EXAMPLE OF TRANSFORMATIONS OF THE PARENT FUNCTION OF SINE:
State the transformations, find the period & amplitude, graph the function over one complete period, determine
the significant points, and give the domain and range of the portion of the graph.
𝑦 = 2 sin(𝑥) − 3 Transformation(s): vertical stretch by a factor of 2,
vertical shift down 3
Period: [0 , 2𝜋] Amplitude: 1 Domain: [0 , 2𝜋] Range: [−5 , −1]
Significant points: (0 , −3), (𝜋2⁄ , −1), (𝜋 , −3), (3𝜋
2⁄ , −5), (2𝜋 , −3)
(0, 0) (𝜋2⁄ , 1) (𝜋 , 0) (3𝜋
2⁄ , −1) (2𝜋 , 0)
∗ 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2
(0, 0) (𝜋2⁄ , 2) (𝜋 , 0) (3𝜋
2⁄ , −2) (2𝜋 , 0)
−3 − 3 − 3 − 3 − 3
(0, −3) (𝜋2⁄ , −1) (𝜋 , −3) (3𝜋
2⁄ , −5) (2𝜋 , −3)
Work to find
the period
Graph of 𝑦 = 2 sin(𝑥) − 3
from previous page
You Try:
𝑓(𝑥) = sin1
2(𝑥 + 𝜋)
Transformation(s): Horizontal stretch by a factor of 2,horizontal shift left 𝜋
Period: _______ Amplitude: _______ Domain: _______ Range: _______
Significant points: _____________________________________________
(0, 0) (𝜋2⁄ , 1) (𝜋 , 0) (3𝜋
2⁄ , −1) (2𝜋 , 0)
𝑔(𝑥) = sin(2𝑥)
Transformation(s): Horizontal stretch by a factor of 2,horizontal shift left 𝜋
Period: _______ Amplitude: _______ Domain: _______ Range: _______
Significant points: _____________________________________________
(0, 0) (𝜋2⁄ , 1) (𝜋 , 0) (3𝜋
2⁄ , −1) (2𝜋 , 0)
Name _______________________________________________ Date ________________________ Block ________
THE SINE FUNCTION ASSIGNMENT
State the transformations, find the period & amplitude, graph the function over one complete period, determine the
significant points, and give the domain and range of the portion of the graph. Clearly show the work and the values
for the significant points.
1. 𝑦 = sin 𝑥 + 2 Transformation(s): ______________________________
_____________________________________________
Period: _______ Amplitude: _______
Domain: _______ Range: _______
Significant points:
_____________________________________________
2. ℎ(𝑥) = sin(𝑥 − 𝜋) Transformation(s): ______________________________
_____________________________________________
Period: _______ Amplitude: _______
Domain: _______ Range: _______
Significant points:
_____________________________________________
3. 𝑔(𝑥) =1
2sin(
1
2𝑥) Transformation(s): ______________________________
_____________________________________________
Period: _______ Amplitude: _______
Domain: _______ Range: _______
Significant points:
_____________________________________________
4. 𝑦 = sin 2(𝑥 +𝜋
2) Transformation(s): ______________________________
_____________________________________________
Period: _______ Amplitude: _______
Domain: _______ Range: _______
Significant points:
_____________________________________________
Sketch the angle and then evaluate the trig function. Show work where necessary.
5. sin 420° _______ 6. cos (−3𝜋
4 ) _______
7. tan11𝜋
6 _______ 8. cos 135° _______
2𝜋
𝑏
3
Set 𝜋 equal to 2𝜋
𝑏
and solve for 𝑏
2
𝜋
4
−1
Once you have the values
for 𝑎, 𝑏, 𝑐, and 𝑑, plug them
into the standard form for
the sine function.
Name __________________________________________________ Date ____________________ Block _________
Phase III – 5/11/2020 to 5/15/2020 – Analyzing Sine Functions Notes and Assignment
To write the equation for a sine function in standard form: 𝑦 = 𝑎 sin b(𝑥 − 𝑐) + 𝑑
1. Determine the midline . Add the minimum & maximum and divide by 2. This is the
vertical shift (𝑑).
2. Find the amplitude (a). This is the difference between the midline value and the max .
3. Find the period (b). This is the difference between any two sets of repeating points.
4. Find the phase shift (c). This is the difference between the 𝑥-value of the first
significant point and (0, 0).
EXAMPLE 1:
EQUATION:
𝑦 = 3 sin 2 (𝑥 −𝜋
4) − 1
𝑎 = ______ 𝑏 = _______ 𝑐 = _______ 𝑑 = _______
Max: 2 Min: −4 |0 − 𝜋| = |−𝜋| = 𝜋 𝜋
4− 0 =
𝜋
4 𝑑 = Midline value
Midline: 2+(−4)
2= −1
amplitude:
|𝑀𝑖𝑑𝑙𝑖𝑛𝑒 − 𝑀𝑎𝑥| = |−1 − 2| 𝑏 ⦁ 𝜋 =2𝜋
𝑏 ⦁ 𝑏
= |−3| 𝑏𝜋 = 2𝜋
= 3 𝜋 𝑏 = 2
Midline
Repeating points
First significant
point: (𝜋
4 , −1)
0 −𝜋
3
1
You try: Write the equation for each sine function. I have
completed some sections for you.
EQUATION: _____________________________
𝑎 = ______ 𝑏 = _______ 𝑐 = _______ 𝑑 = _______
−𝜋
3− 0 = −
𝜋
3 Midline =
2+(−2)
2= 0
Equation: ________________________
𝑎 = ______ 𝑏 = _______ 𝑐 = _______ 𝑑 = _______
|−𝜋
4−
7𝜋
4| = |−
8𝜋
4| = 2𝜋
𝑏 ⦁ 2𝜋 =2𝜋
𝑏 ⦁ 𝑏
2𝜋𝑏 = 2𝜋
2𝜋 2𝜋 𝑏 = 1
Midline
First significant
point: (−𝜋
3 , 0)
Repeating points
Name _______________________________________________ Date ________________________ Block ________
ANALYZING SINE FUNCTIONS ASSIGNMENT
Given the sine functions, identify the transformations, period, and amplitude. Clearly show work to find the
significant points. Sketch the graph over one complete period on a clearly labeled grid and identify the domain and
range.
1. 𝑔(𝑥) = 2 sin 𝑥 − 4 Transformation(s): ______________________________
_____________________________________________
Period: _______ Amplitude: _______
Domain: _______ Range: _______
Significant points:
_____________________________________________
2. 𝑓(𝑥) =1
2 sin 2(𝑥 − 𝜋) Transformation(s): ______________________________
_____________________________________________
Period: _______ Amplitude: _______
Domain: _______ Range: _______
Significant points:
_____________________________________________
Sketch the angle and then evaluate the trig function. Show work where necessary.
3. sin 315° _______ 4. tan (−5𝜋
4) _______ 5. cos
11𝜋
3 _______
Give the equation of the sine curve.
6. Equation: ________________________
𝑎 = ______ 𝑐 = _______
𝑏 = _______ 𝑑 = _______
7. Equation: ________________________
𝑎 = ______ 𝑐 = _______
𝑏 = _______ 𝑑 = _______
8. Equation: ________________________
𝑎 = ______ 𝑐 = _______
𝑏 = _______ 𝑑 = _______
