Transforming the Sine and Cosine Function

Sine Graph vs. Cosine Graph

y= asin[b(x - h)] + k

y= acos[b(x - h)] + k

BOTH ARE AFFECTED BY THE SAME VARIABLES

a= amplitude (half the distance between the highest and lowest points)<- when this is negative the graphs flip

b= 2π/period

Period= one cycle of sine or cosine

h= horizontal shift 

k= vertical shift

BOTH GRAPHS AFFECTED BY THE SAME VARIABLES

<- POSITIVE SIN GRAPH<-POSITIVE COSINE GRAPH

Questions:

1) Find the equation for the following graph:

2) Find the following equation:

3) y=2sin (1/2(x-2))+1

a. Find the amplitude

b. Find the period

c. Find the vertical stretch

d. Find the horizontal stretch

4) How is this equation different from the parent of cosine graph?

y=-1/2 cos (2(x+pi/3))+1

5) What does an amplitude of 2 mean on the unit circle?

Click here for answers :)

Unit Circle

Unit Circle:
What is the unit circle?
Click here to explore an interactive unit circle.
A unit circle is a circle with the radius of 1 which helps determine the sin, cos, tan of given radians. The unit circle is an easy way to look at angles using radians.

The unit circle we can help us determine sin, cos, and tan of each given radian or degree.
Lets take a simple example of π/4 (45 degrees)
On the unit circle 45 degrees is written as π/4.
Because the angle is 45 we can conclude this forms a special right triangle!

The triangle on the right is what is formed from this radian
and gives us the ability to find sin, cos, and tan.

Remembering our SOHCAHTOA, we can use this triangle to determine the sin, cos, and tan.
Find sine:

Since sin= opposite of θ/hypotenuse
sin= 1/√ 2 = √ 2 / 2  (this is the height)
Find cosine:
Since cos=adjacent of θ/hypotenuse
cos=1/√ 2 = √ 2 / 2 (this is the base length)
Find tangent:
Since tan=opposite of θ/adjacent of θ
tan=1/1= 1 

An easy way to find the cos and sin is using the Pythageorean Identity
(sin)^2+ (cos)^2= 1
so...
If sin= 1/2, what is cos?
(1/2)^2+cos^2=1
1-1/4=cos^2
3/4=cos^2 (SQUARE both sides)
to get cos=  √3/2

<-SHOWS THE SIN,COS

Problems:

1) Find sinπ/3 on the unit circle, what does it mean?

2) If cosine= √ 2/2, what is sin?

3) What is the tangent of 120 degrees?

4) Show the meaning of 3 radian on the unit circle.

5) What is the sin of π/2? 

6)Why is π/4 equal to 45 degrees?

Click for answers :)

Radian and Transforming Radian to Degrees

What is Radian?
Just like a degree, radian is used to measure angles.
Radian is the number the radius is in a circle which is 2π= 6.283 (This is true for ALL CIRCLES)

By using a simple equation you can easily turn radian into degrees!
Formula:  2π= r x 360/ degrees
ex. 45° into Radian?
Step 1) Write the equation: 2π= r x 360/ degrees
Step 2) Put given information into equation: 2π= r x 360/ 45°
Step 3) Solve for r: 
2π= r x 360/ 45° (divide 360r/45)
2π= 8r 
2π/8= 8r/8 (divide 8 on both sides)
2π/8= r (simplify answer to get...)
45°=π/4
To further understand Radians lets take a look at the Unit Circle...


Problems:
1) Transform the following radian to degree and degree to radian:

a. 60° 

b. 430°

c. π/10

d. 11π/12

Click here for answers

Answers: Transforming the Sine and Cosine Function

1) y= cos (1/2x)

2)  y=-cos (2x)

3) y=2sin (1/2(x-2))+1

a. amplitude= 2

b. period=4pi

c. 2

d. 1

4) Amplitude is 1/2, the graph is moved 1 up and right by pi/3. The period of this graph is pi.

5) The length of the radius

Answers: Radian and Transforming Radian to Degrees

1)
a. 60° = π/3



b. 420°=7π/3 rad



c. π/10= 18 degrees




d. 11π/12=165 degrees

Answers: Unit Circle

1) 60 degree angle of rotation

2) √ 2/2, what is sin? Sin=√ 2/2

3) 1

4) SEE BELOW RADIAN CIRCLE

5) 1

6) 180/4= 45