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

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)

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

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

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

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

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

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

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

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

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)

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