C 1000 a Identify the amplitude of f b Identify the period o

C) 1000. a. Identify the amplitude of f b. Identify the period off c. Identify the phase shift of f d Identify the midline off

10 . f(x)= 4sin(2(x-2)) + 12

= 4sin(2x-4)+12

This equation is of the form f(x)= Asin(Bx-C)+D

Amplitude ,A= 4

Period=2pi/B=2pi/2=pi

Phase shift=C/B= 4/2=2

6. tan(theta/2)=csc theta -cot theta

starting with the left side

tan(theta/2)= (1-costheta)/sin theta=1/sin theta - costheta/sin theta=csc theta-cot theta

b. (cos theta - sin theta)²=1-sin 2theta

starting with the left side

(cos theta - sin theta)²=cos²theta + sin²theta-2sin theta cos theta

and cos²theta + sin²theta=1 ,2sin thetacos theta=sin 2 theta

Therefore we get

1+ sin 2 theta

c. sin (theta/2)=sin theta/(sqrt2(1+cos theta))

using half angle identity

sin(theta/2)= sqrt((1-cos theta)/2)

Multiplying top and bottom by 1+cos theta inside the square root

sqrt((1-cos theta)(1+cos theta)/2(1+cos theta))= sqrt(sin²theta/2(1+cos theta))

= sin theta/sqrt(2(1+cos theta))