Harmonic Form

Question 1

What harmonic form do you use for aSinx + bCosx ?

Answer 1

RSin(x + α)

Question 2

What harmonic form do you use for aCosx + bSinx ?

Answer 2

RCos(x - α)

Question 3

What harmonic form do you use for aSinx - bCosx ?

Answer 3

RSin(x - α)

Question 4

What harmonic form do you use for aCosx - bSinx ?

Answer 4

RCos(x + α)

Question 5

When do you use RSin(x + α) ?

Answer 5

aSinx + bCosx

Question 6

When do you use RCos(x - α) ?

Answer 6

aCosx + bSinx

Question 7

When do you use RSin(x - α) ?

Answer 7

aSinx - bCosx

Question 8

When do you use RCos(x + α) ?

Answer 8

aCosx - bSinx

Question 9

What angle unit should you always assume for harmonic form, if not mentioned in the question?

Answer 9

radians

Question 10

Write 5Cosθ - 12Sinθ in harmonic form

Answer 10

Let 5Cosθ - 12Sinθ ≡ RCos(θ + α)
≡ R(CosθCosα - SinθSinα)
≡ RCosθCosα - RSinθSinα

Sinθ I -12=-Sinα –> 12=RSinα
CosθI 5=RCosα
R = √ (12^2) + (5^2) = 13

RSinα/RCosα = tanα
tan α = 12/5 –> tan-1(12/5) = 1.18 –>α=1.18

5Cosθ - 12Sinθ ≡ 13Cos(θ + 1.18)

Question 11

(minimum points)
How do you make the expression as small as possible?

Answer 11

make the denominator as big as possible

Question 12

How do you state the minimum value?

Answer 12

r x -1

Question 13

How do you state the maximum value?

Answer 13

r x 1

Question 14

The harmonic form of 3Cosx - 2Sinx is √13Cos(x + 33.7), state it’s minimum value and the value of x at which it occurs.

Answer 14

minimum value = √13 x -1 = -√13
Occurs when Cos(x+33.7) = -1
x+33.7 = 180
x=146.3

to find when maximum value occurs, equal it to 1

Question 15

What does [ mean?

Answer 15

including

Question 16

What does ) mean?

Answer 16

up to

Question 17

How would you rewrite the interval [0, π) ?

Answer 17

0≤ θ<π