PreCalc Chapter 5 Review

Identities Fill-In-The-Blank (Part 1)

100

Complete the following identity:

sin^2x+cos^2x=?

sin^2x+cos^2x=1

200

Complete the following identity:

1+?=sec^2theta

1+tan^2theta=sec^2theta

300

Complete the following identity:

1+cot^2u=?

1+cot^2u=csc^2u

400

Complete the following identity:

tan(pi/2-x)=?

tan(pi/2-x)=cotx

500

Complete the following identity:

tan(-u)=?

tan(-u)=-tanu

Solve for x (Part 2)

100

Find all exact solutions to the equation in the interval  [0,2pi) 

2sinx+sqrt2=0

sinx=-frac{sqrt2}{2}

x=frac{5pi}{4},frac{7pi}{4}

200

Find all exact solutions to the equation in the interval  [0,2pi) 

tanxsecx-2tanx=0

tanx(secx-2)=0

tanx=0->x=0,pi

secx=2->cosx=1/2->x=pi/3,frac{5pi}{3}

300

Find all exact solutions to the equation in the interval  [0,2pi) 

7sin10x+7=0

sin10x=-1

10x=frac{3pi}{2}

x=frac{3pi}{20}

400

Find all exact solutions to the equation in the interval  [0,2pi) 

sin2x+2cosx=0

2sinxcosx+2cosx=0

2cosx(sinx+1)=0

2cosx=0->x=pi/2,frac{3pi}{2}

sinx+1=0->sinx=-1->x=frac{3pi}{2}

500

Find all exact solutions to the equation in the interval  [0,2pi) 

8cos^2x-10cosx=7

8cos^2x-10cosx-7=0

(4cosx-7)(2cosx+1)=0

cosx=7/4->no solution

cosx=-1/2->x=frac{2pi}{3},frac{4pi}{3}

Given Formulas (Part 2)

100

Evaluate the exact value of  sinfrac{7pi}{12}.  Use the fact that  frac{7pi}{12}=frac{4pi}{3}-frac{3pi}{4} 

sinfrac{4pi}{3}cdotcosfrac{3pi}{4}-cosfrac{4pi}{3}cdot sinfrac{3pi}{4}

-frac{sqrt3}{2}cdot -frac{sqrt2}{2}- -1/2cdot frac{sqrt2}{2}

=frac{sqrt6+sqrt2}{4}

200

Evaluate the exact value of  cos165º using an appropriate sum/difference formula.

cos(120º+45º)

cos(frac{2pi}{3}+pi/4)

cosfrac{2pi}{3}cdotcosfrac{pi}{4}-sinfrac{2pi}{3}cdot sinfrac{pi}{4}

-1/2cdot sqrt2/2 - sqrt3/2 cdot sqrt2/2

frac{-sqrt2-sqrt6}{4}

300

Simplify the expression  sin(pi-x) using an appropriate sum/difference formula.

sinpicdotcosx-cospicdot sinx

0cdotcosx- -1cdot sinx

sinx

400

Given  sintheta=4/5 and  costheta<0 , find  cos2theta .

(Draw a triangle)

cos2theta=cos^2theta-sin^2theta

=(-3/5)^2-(4/5)^2

=-7/25

500

Find  cosfrac{3pi}{8} . Use the fact that  frac{3pi}{8}=1/2(frac{3pi}{4}) 

cosfrac{u}{2}=+-sqrt{frac{1+cosu}{2}}

cosfrac{frac{3pi}{4}}{2}=sqrt{frac{1+cosfrac{3pi}{4}}{2}}

=sqrt{frac{1+ -sqrt2/2}{2}}

=sqrt{frac{2-sqrt2}{4}}

=frac{sqrt{2-sqrt2}}{2}