Lesson 29 - y=asin( Ѳ - p) + C

y=cosѲ → 2 cosѲ   2 in front means there is a vertical expansion of 2

y= cosѲ  → cosѲ +1     +1 means the graph will move up  1 on the center line (vertical displacement)

y=1/4sinѲ   ¼ treat as 1’s on your y-axis, instead of 1 and -1 it will be ¼ and -1/4 

y= sin( Ѳ + π/2)    ← π/2  this is called a phase shift, it will move to the left π/2  

y= cos (Ѳ - π/3)     

→ π/3    1/4P = π/2    Phase Shift: = π/3 = 2 π/6  Scale: (common denominator) π/6    

 

Lesson 28- y= SinѲ, CosѲ, TanѲ

Repciprocal Trig Functions:

Cosecant: cscѲ= 1/ sin Ѳ
Secant: secѲ= 1/cosѲ
Cotan: tan Ѳ = 1/tanѲ

Examples:
csc 2π/3 = 1/ sin(2π/3)  * 2π/3 = 2/√3 and is then flipped with its reciprocal √3/2

sec 3π/2 = 1/ cos( 3π/2)  * 3π/2=  Ø

cot30 = 1/tan30= -0.16

Solve:  2cos Ѳ - √3 = 0       [ 0, 2π]
           2cosѲ= √3       - isolate cos
               2       2
            cosѲ = √3         Ѳ= π/6 
                        2          Ѳ2=11 π/6

Sin, Tan, Cos Graphs:

CosѲ graph:  starts on top, ends at top, makes a U shape, has a period of 2π(360°) and a quarter period of  π/2(90°)   

SinѲ graph: starts on mid-line, goes up, back to mid-line then down, and repeats. Making a squiggly line, has to the same period of cos
TanѲ graph: has a asymptote at 90°( 2π/4) and another every 180°, tan has a period of π, the y-scale goes from infinity to negitive infinity

  


                     


 

Lesson 27- Special Angles

Equilateral Triangle:

30°, 60°, 90°      1: (square root)3: 2

Iscolese Triangle:

45°, 45°, 90 ° 

	0°	30°	45°	60°	90°	120°	135°	150°
CosѲ	√2/3	√3/2	√2/2	√1/2	√0/2	-√1/2	-√2/2	-3/2
SinѲ	√0/2	1/2	√2/2	√3/2	√1/2	√3/2	√2/2	√0/√2
TanѲ	0	√3/1	1	√3/1	Ø	-√3	-1	-√3/3

Lessson 26- SPA

Standard Position Angle: SPA

SinѲ= y/1 = y  (y/r)
CosѲ= x/r = x/1 = x
TanѲ= y/x = rise/run = slope

Solve  0° <  Ѳ  < 360°     SinѲ=  - 0.5641

Step 1: Draw SPA

Step 2: Use Calc  2nd sin -0.5641)

Step 3: Use symmetry to find answers