Pre-Calculus 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 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)

300

If csc x = sqrt(15)/3, find sin x exactly.

csc x = 1/sinx

1/sin x = sqrt(15)/3

sin x = 3/sqrt(15) = sqrt(15)/5

400

The exact value of sin(240^o).

sqrt(3) / 2

500

If the exact value of cos(x) is 5/7, and tan x < 0, find all five other trig ratios exactly.

sin x = -2sqrt(6)/7

sec x = 7/5

csc x = -7sqrt(6)/12

tan x = -2sqrt(6)/5

cot x = -5sqrt(6)12

Verifying/Simplifying Trig. Identities

100

sec²x(1 - cos²x)

sec²x(1 - cos²x)

= sec²x - (sec²x)(cos²x)

= sec²x - (1/cos²x)(cos²x) 

= sec²x - 1 

= tan²x 

200

(cot²x)(csc²x) - cot²x = cot⁴x

(cot²x)(csc²x) - cot²x

= (cot²x)(csc²x - 1)

= (cot²x)(cot²x)

= cot⁴x

300

How is the verification of identities different than solving trig equations?

Verification you are working with one side to manipulate it to look like the other side.

Solving equations you are still kind of manipulating but you can work with both sides, and there are specific answers you are looking for.

400

PICK A NUMBER (Between 0-1000)

Gain that many points

500

csc x sec x - tan x

csc x sec x - tan x 

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

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

= 1/(sinxcosx) - sin²x/(sinxcosx)

= (1-sin²x)/(sinxcosx)

= (cos²x)/(sinxcosx)

= (cosx/sinx)

= cotx

Trig Equations

100

Find all the solutions in degrees between 0 and 360:

4=-8cos(x)

-1/2 = cos x

x = 120 and 240

200

Find all the solutions on the unit circle in degrees between 0 and 360:

5=1+2csc(x)

5 = 1 + 2csc(x)

4 = 2csc(x)

2 = csc(x)

1/2 = sin(x)

x = 30 and 150

300

PICK A NUMBER (between 0 and 1000)

Lose that many points.

400

Find all radian solutions [0,2pi) of 

2cos²x + cosx - 1 = 0

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

2cosx + 1 = 0  and cosx - 1 = 0

cosx = -1/2  and   cosx = 1

x = 2pi/3, 5pi/3, 0, pi

500

Solve for all angles between 0 and 2pi:

(cos x)(sin x) - (sin x) = 0

(sin x)(cos x - 1) = 0

sin x = 0 and cos x - 1 =0

x = 0, pi and x = pi/2

Solving Trig. Equations

100

Solve for all angles between 0 and 2pi.

5sin x+2=sin x

5sinx+2 = sin x

4sinx+2=0

2(2sinx+1)=0

2sinx = -1

sin x = (1/2)

x = 7pi/6 and 11pi/6 

200

Find all solution on the unit circle between 0 and 2pi.

cot x = sqrt(3)

cot x = sqrt(3)

tan x = 1/sqrt(3)

tan x = sqrt(3)/3

x = pi/6, 7pi/6

300

Find all solutions for x.

2-10secx=4-9secx

2-10secx = 4-9secx

-10secx = 2-9secx

-secx = 2

secx = -2 

cosx = -(1/2)

x = 2pi/3 + 2npi, 4pi/3 + 2npi, n = int.

400

Find all solutions for x.

3cscx = 2cscx+(√2)

3cscx = 2cscx+(√2)

cscx = (√2)

sinx = (1/(√2))*((√2)/(√2))

sinx = (√2)/2)

x = pi/4 +2npi, 3pi/4 + 2npi, n = int.

500

Find all solutions for x.

11=3csc²x+7

11=3csc²x+7

4=3csc²x

(4/3) = csc²x

 (+/-)((√(4/3))=(√csc²x)

csc x = (+/-)(2/(√3))or(+/-)(2(√3)/3)

sin x=(+/-)(√3/2)

x = (pi/3), (2pi/3), (4pi/3), (5pi/3) all + 2npi, n = int.

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

This slow and slimy animal has four noses.

What is a slug?

Snapple fact #3.

500

This is Barbie's full name

What is Barbara Millicent Roberts?  Snapple Fact #902