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Sum and Difference Identities
Polar Graphs
Vectors
Complex Numbers
Bad Korean Jokes
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

What type of graph is r = 2 - 3 cos(θ)?

Limacon with loop (because |b| > |a|)
100
What is the sum of vector u if u = <2, -1> and 2v if v = <5, 4>?
2v = <10, 8>

u + 2v = <12, 7>

100
What is the modulus of the complex number 3 + 4i?
Mod = sqrt(9 + 16) = 5
100

This is what you do in your free time while studying computer science overseas.

유학!

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

What type of polar graph is r = 4 cos(θ)?

Circle, radius = 2, center at (2, 0)
200
What are the magnitude and argument of vector y = <9, 12> to the nearest degree?

Magnitude: sqrt(81 + 144) = 15

Argument: tan-1(12/9) = 53 degrees

200
Convert the complex number 3 + 4i into polar form.

r = 5, θ = 53 degrees

-> 5 (cos 53 + i sin 53) or 5 cis (53)

200

Which body part gets the best workout whenever you go really, really fast?

허리

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

What type of polar graph is r = cos(4θ)?

Rose curve with 8 petals
300
What is the dot product of vectors u = <2, -1> and v = <5, 4>?
Dot product = 2x5 + -1x4 = 10 - 4 = 6
300
Find the product of z = 3 + 4i and x = 2 - 2i in polar form.

|z| = 5, θ1 = 53, |x| = sqrt(8), θ2 = 315

zx = 5 sqrt(8) cis(368) or 10 sqrt(2) cis(368)

= 10sqrt(2) cis(8)

300

What happens when a romantic couple eats fruit together?

They have 반하나.

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

What type of polar graph is r = 5?

Circle, radius = 5, center at (0, 0)
400
What is the angle between the unit vectors i = <1, 0> and j = <0, 1>?
Since the dot product is 0, they are orthogonal.

Therefore, the angle is 90 degrees.

400

Find the rectangular form of (3 + 4i)9

3+4i = 5 cis(53.13)

(5 cis(53.13))9 = 59 cis(9x53.13)

= 1,953,125 cis 478.17

= 1,953,125 (cos 118.17 + i sin 118.17)

= -922049.32 + 1,721,778.82 i

400

What do you call the countries who always have a good reason for everything?

The 이유.

500

Use sin(x+y) = sinx cosy + cosx siny to prove that sin(2x) = 2 sin(x) cos(x)

sin(x+x) = sinx cosx + cosx sinx

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

sin(2x) = 2 sin x cos x

500

What type of polar graph is r2 = 4 cos(2θ)?

Lemniscate symmetric about polar axis
500
What is the angle between vectors u <0, 5> and v <6, 4>?

Dot product = 0 + 20 = 20

Magnitude of u = 5, magnitude of v = sqrt(52)

20 = 5 sqrt(52) cos(θ)

cos(θ) = 0.5547

θ = 56.3 degrees

500
Find the two square roots of i.

r = 1, θ = 90 degrees

Principal root: (1 cis 90)1/2

= 1 cis(1/2 x 90) = cis(45)

= sqrt(2)/2 + sqrt(2)/2 i

Secondary root: (1 cis(90+360))1/2

= (1 cis(450/2)

= 1 cis(225) = -sqrt(2)/2 - sqrt(2)/2 i

500

Which part of your body is always outside?

Your 밖.