1. By: Advay Muchoor, Atsushi Hikawa, and Anjali Singh
Unit 12
By: Advay Muchoor, Atsushi Hikawa, and Anjali Singh

2. 13.4 Evaluate Inverse Trig Functions
y=sin x restrictions
Domain: -π/2≤x≤π/2
Range: -1≤x≤1
y=cos x restrictions
Domain: 0≤x≤π
y=tan x restrictions
Range: ARN
y=arcsin x restrictions
Domain: -1≤x≤1
Range: -π/2≤x≤π/2
y=arccos x restrictions
Range: 0≤x≤π
y=arctan x restrictions
Domain: ARN
Remember ASTC (All Students Take Calculus) --- Starting from quadrant I, the positive identities are All, Sine, Tangent, and Cosine)

3. Practice
sin-11
tan-1(-√3/3)
cos-1(-2)
90°(π/2)
-30°(-π/6)
no solution

4. 14.3 Verify Trig Identities
Reciprocal Identities
csc x= 1/sin x
sec x= 1/cos x
cot x= 1/tan x
Tangent/Cotangent
tan x= sin x/cos x
cot x= cos x/sin x
Negative Angle Identities
sin (-x)= -sin x
tan (-x)= -tanx
cos (-x)= cos x
Pythagorean Identities
sin2x+cos2x=tan2x
1+tan2x=sec2x
1+cot2x=csc2x
Cofunction Identities
sin((π/2)-x)=cos x
cos((π/2)-x)=sin x
tan((π/2)-x)=cot x
sec((π/2)-x)=csc x

5. Verify Identities Practice
Verify the identity: secΘ-cosΘ = sinΘtanΘ

6. 14.4 Solve Trig Equations EX: 2cos2x+1=2 in the interval 0≤x≤2π
**Trigonometric Identities are true for all real values of Θ
**Trigonometric Equations are true for one or more values of Θ
EX: 2cos2x+1=2 in the interval 0≤x≤2π
2cos2x=1
cos2x=½
cosx=±√2/2
x=π/4,3π/4,5π/4,7π/4

7. 14.5 Write Trig Functions and Models
Find max (M) and min (m).
Identify the “k” value. k= (M+m)/2
Decide if graph is sin or cos, and then find the “h” value.
Find the amplitude and period: |a|= (M-m)/2

8. Practice
Write the equation for this graph:
y= 2 sin 1/3 (x-(π/6))

9. Connection to Other Units
This unit is a lot like unit 4, Quadratic Functions and Factoring
Just like this unit, using an “h” and a “k” affected the parent graph of y=x2
For example: y=(x-4)2+3 has an h value of 4 and a k of value of 3
That shifts the graph just like sinusoids
In sinusoids, the k value moves it up or down just like quadratic equations. The h value also shifts it to the right or left

10. Real Life Examples Music Astronomy Cell phones and GPS(navigation)
Electrical Currents
Architecture
Geology
Etc.

11. Common Mistakes Doing the vertical shift before the horizontal shift
Mixing up sine and cosine when writing equations
Forgetting what functions are positive or negative in different quadrants (ASTC)
Silly mistakes in messy work (bad handwriting)

12. IN CONCLUSION
YOU HAVE A LOT TO STUDY!!!!