Trig-pardy

Trig. Identities

100

If cos x = (2/3) find sec x

What is sec x = (1/cos x)

=(1/(2/3))

=(3/2)

200

If tan a = (1/5) find cot a

What is cot a = (1/tan a )

=(1/(1/5))

=(5/1)

=5

300

if cos x = (1/6) and sin x = ((√35)/6) find cot x

What is cot x = ((cos x)/(sin x))

=((1/6)/(√35)/6)

=(1/(√35)) * ((√35)/(√35))

=((√35)/35)

400

csc x - cos x cot x

What is csc x-cosx cot x = (1/sinx)-cosx(cosx/sinx)

=(1/sin x)-((cos²x)/(sinx))

=((1-cos²x)/(sinx))

=((sin²x)/(sinx))

=sinx

500

csc x sec x - tan x

What is csc x sec x - tan x = (1/sinx)*(1/sinx)-(sinx/cosx)

=(1/(sinxcosx))-(sinx/cosx)

=(1/(sinxcosx))-(sinx/sinx)(sinx/cosx)

=((1-sin²x)/(sinxcosx)

=((cos²x)/(sinxcosx)

=(cosx/sinx)

=cotx

Verifying/Simplifying Trig. Identities

100

(sec²x-1)cos²x

What is (sec²x-1)cos²x

=(tan²x)cos²x    Pythagorean Identity

=((sin²x)/(cos²x))cos²x  Quotient Identity

=sin²x   Multiply and divide out common factor

200

sec²x(1-cos²x)

What is sec²x(1-cos²x)

=sec²x-sec²xcos²x   Distributive Property

=sec²x-(1/(cos²x))*cos²x Reciprocal Identity

=sec²x-1 Multiply and divide out common factor

=tan²x Pythagorean Identity

300

sinx-sinx cos²x=sin³x

What is sinx-sinx cos²x

=sinx(1-cos²x)  Factor

=sinxsin²x  Pythagorean Identity

=sin³x Multiply

400

cot²x csc²x-cot²x=cot⁴x

What is cot²xcsc²x-cot²x

=cot²(csc²x-1)  Factor

=cot²xcot²x  Pythagorean Identity

=cot⁴x Multiply and add exponents 

500

tanx csc²x-tanx

What is tanx csc²x-tanx

=tanx(csc²x-1) Factor

=tanx cot²x Pythagorean Identity

=((sinx)/(cosx))*((cos²x)/(sin²x)) Quotient Identities

=(cosx)/(sinx)  Multiply and divide common

=cot x quotient identity

Trig Equations

100

Find the solutions on the unit circle in degrees:

4=-8cos(x)

120 and 240

200

Find the solutions on the unit circle in degrees:

5=1+2csc(x)

30 and 150

300

Find the solutions on the unit circle in radians:

-4=-2+2tan(x)

3pi/4 and 7pi/4

400

Find the solutions on the unit circle in radians:

-csc(x)+cot(x)=-1

pi/2

500

Find the solutions on the unit circle in degrees:

0=-2cot(x)-csc²(x)

135 and 315

Solving Trig. Equations

100

5sinx+2=sinx

What is 5 sinx+2=sinx

=4sinx+2=0

=2(2sinx+1)=0

=2sinx=-1

= sinx=(1/2)  The period of sine is (2pi), so you only need to find solutions on the interval [0,(2pi)].  The solutions on this interval are ((7pi)/6) and ((11pi)/6). 

200

2-10secx=4-9secx

What is 2-10secx=4-9secx

=-10secx=2-9secx

=-secx=2

=secx=-2  = cosx=-(1/2)

((2pi)/3)+2npi, ((4pi)/3)+2npi, neZ

300

5=sec²x+3

What is 5=sec²x+3

= 2=sec²x

=(+/-)(√2)=(√sec²x)

= secx=(+/-)(√2)

= cosx=(+/-)(1/(√2))*((√2)/(√2))

= cosx=(+/-)((√2)/2)

=(pi/4),(3pi/4),(5pi/4),(7pi/4)

400

3cscx=2cscx+(√2)

What is 3cscx=2cscx+(√2)

= cscx=(√2)

= sinx=(1/(√2))*((√2)/(√2))

= sinx=(√2)/2)

=(pi/4), ((3pi)/4)

500

11=3csc²x+7

What is 11=3csc²x+7

= 4=3csc²x

=(4/3) = csc²x

= (+/-)((√(4/3))=(√csc²x)

=cscx = (+/-)(2/(√3))or(+/-)(2(√3)/3)

= sinx=(+/-)(√3/2)

= (pi/3), (2pi/3), (4pi/3), (5pi/3)

Snapple Facts

100

This island is the largest in the Continental U.S.

What is Long Island? Snapple Fact #333

200

This letter was the last letter to be added to the English alphabet 

What is J?  Snapple Fact #336

300

This aquatic mammal sleeps with one eye open

What is a dolphin? Snapple Fact #760

400

Playing ping-pong for 12 hours will burn this much weight

What is 1 pound?  Snapple Fact #167

500

This is Barbie's full name

What is Barbara Millicent Roberts?  Snapple Fact #902