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Exp/Log Equations
Identities
Sum/Difference
Trig Equations
Bad Korean Jokes
100

What is a logarithm?

The inverse of an exponential.
100

What common substitution can be made to replace sin2 (x)?

1 - cos2 (x)

100
sin(x+y) = sinx cosy + cosx siny. Using degrees, what is the exact value of sin(165)?
sin(165) = sin(120 + 45)

= sin120 cos45 + cos120 sin45

= sqrt(6)/4 - sqrt(2)/4

= (sqrt 6 - sqrt 2) / 4

100

Why does the following equation have no solutions?

sin2 (x) - 4 = 0

(sin x + 2) (sin x - 2) = 0

sin x = -2 and sin x = 2 are both outside of sine's range and therefore have no solutions.

100
Which part of Korea is the most competitive in sports matches?
"경기"-도
200

2x=24. What is the value of x?

x = 4.585
200

What is a shorter replacement for tan2 (x) + 1?

sec2 (x)

200
Given cos(x-y) = cosx cosy + sinx siny, find the exact value of cos(-15) degrees.
cos(-15) = cos (45 - 60)

= cos45 cos60 + sin45 sin60

= sqrt(2)/4 + sqrt(6)/4

= (sqrt 2 + sqrt 6)/4

200

2 cos2 (x) = cos x + 1. Solve for x between [0, 360) in degrees.

x = 0, 120, 240
200
What do you call the rain that soaks all college students?
학비
300

Solve: log3 ⁡(243)= x

x = 5
300

Using cos(B) = x/r and sin(B) = y/r, write tan(B) in terms of these variables.

tan(B) = y/x
300
Given two quadrant 1 angles such that sin A = 0.5, sin B = 0, and sin(A-B) = sinA cosB - cosA sinB, find sin(A-B).
cos A = sqrt(3)/2 and cos B = 1

(1/2) (1) - (sqrt(3)/2) (0) = 1/2 = 0.5

300

Solve 3 tan2(x) + 1 = 0 for x using radians.

x = π/6, 5π/6, 7π/6, 11π/6

300
Which part of Seoul gives the friendliest greetings?
인사-동
400

A population is modeled by the function P = 5000 (1.01)x, where P is population and x is years. At this rate, how many years will pass before the population reaches 6000 people?

About 18.32 years.
400

Prove that tan(x) / cot(x) = sec2 (x) - 1

tan(x)/cot(x) = sec2(x) - 1

tan2(x) = sec2(x) - 1

tan2(x) = tan2(x)

Fin!

400
Using cos(x+y) = cosx cosy - sinx siny, cos A = 3/5, cos B = 24/25, find cos(A+B).
sinA = 4/5, sinB = 7/25

(3/5)(24/25) - (4/5)(7/25)

= 72/125 - 28/125

= 44/125

400

Given tan^2(x) + 1 = 1, solve all values of x for [0, 2π).

x = 0, π

400
What is the sweetest object in the night sky?
달
500

Continuously compounded interest is modeled with the formula A = P ert, with P = principal (starting), r = interest rate (as a decimal), t = time (years), A = amount at time t. From investing $1,000 in an account, how long should it take that account to reach $1,100 at a 2% annual interest rate?

4.7655 years.
500

Using cos(x+y) = cosx cosy - sinx siny, prove that cos(2x) = 1 - 2sin2(x)

cos(2x) = cosx cosx - sinx sinx

cos(2x) = cos2(x) - sin2(x)

cos(2x) = 1 - sin2(x) - sin2(x)

cos(2x) = 1 - 2 sin2(x)

500

Use sin(x+y) = sinx cosy + cosx siny to prove that sin(3x) = 3 sin(x) - 4 sin3(x)

sin(3x) = sin(2x)cos(x) + cos(2x)sin(x)

sin(3x) = 2 sin(x) cos2(x) + (sin x)(1-2sin2(x))

sin(3x) = 2sin(x) (1-sin2(x)) + sinx - 2sin3(x)

sin(3x) = 2sinx - 2sin3(x) + sinx - 2sin3(x)

sin(3x) = 3 sin(x) - 4 sin3(x)

500

If -2 cos2(x) - sin(x) + 2 = 0, what is the only possible value of csc(x) and at what two angles between [0, 2π) does it occur?

csc(x) = 2; angles: π/6, 5π/6

500
What kind of gun will always dry you out?
수건