INSTRUCTORS Raffi Hovasapian John Zhu

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 0 answersPost by Kwan Ling Cheung on August 19 at 01:32:27 AMQ10 (i.e. last question) in Practice Questions:What the instructor wants to say in Step 6 is:ADVANCED NOTE: This is the exponential expression of the trigonometric identity(e^(2i*theta) - e^(-2i*theta))/(2i) = sin(2*theta) 0 answersPost by Kwan Ling Cheung on August 19 at 01:14:48 AMQ9 in Practice Questions:The answer should be x = 3 instead of x = 5.Since 3^(x+1) = 81,81 = 9^2 (This step is correct)3^(x+1) = (3^2)^2 ((3^3)^2 is incorrect)3^(x+1) = 3^4x+1 = 4x = 3

### Exponential Functions

• Useful identities:
• Sometimes helpful to think of numbers as exponents: , , etc.

### Exponential Functions

Simplify (x2)3
• This is equivalent to x2x2x2, which would give an exponent of 2 + 2 + 2, or 2 * 3
x6
Rewrite (ex)−2
(ex)−2 = e−2x = [1/(e2x)]
Rewrite (ex)x
(ex)x = ex * x = ex2
Combine the exponents of ([(ex)/(ey)])z
([(ex)/(ey)])z = (ex e−y)z = (ex − y)z = ez(x − y)
Combine the exponents of [(x−5)/(x4)]
[(x−5)/(x4)] = [1/(x4 x5)] = [1/(x4 + 5)] = [1/(x9)]
Simplify [(cos3(x)sin2(x))/(cos5(x)sin3(x) + sin4(x))]
• [(cos3(x)sin2(x))/(cos5(x)sin3(x) + sin4(x))] =
• [(cos3(x)sin2(x))/(sin3(x)(cos5(x) + sin(x)))]=
• [(sin2(x))/(sin3(x))] [(cos3(x))/(cos5(x) + sin(x))] =
• [1/sin(x)] [(cos3(x))/(cos5(x) + sin(x))]=
[(csc(x)cos3(x))/(cos5(x) + sin(x))]
Simplify e−iθe
e−iθe = e−iθ+ iθ = e0 = 1
Solve for x, 2x = 16
• 2x = 16
• 2x = 42
• 2x = (22)2
• 2x = 24
x = 4
Solve for x, 3x + 1 = 81
• 3x + 1 = 81
• 3x + 1 = 92
• 3x + 1 = (33)2
• 3x + 1 = 36
• x + 1 = 6
x = 5
Prove [(2(e − e−iθ)(e + e−iθ))/4i] = [(e2iθ − e−2iθ)/2i]
• [(2(e − e−iθ)(e + e−iθ))/4i]
• =[2/4i] (e − e−iθ)(e + e−iθ)
• = [1/2i] (ee − e−iθe + ee−iθ − e−iθe−iθ)
• = [1/2i] (e2iθ − e0 + e0 − e−2iθ)
• = [(e2iθ − e−2iθ)/2i]
• ADVANCED NOTE: This is the exponential expression of the trig identity 2sin(θ)cos(θ) = sin(2θ)
[(2(e − e−iθ)(e + e−iθ))/4i] =[2/4i] (e − e−iθ)(e + e−iθ)
= [1/2i] (ee − e−iθe + ee−iθ − e−iθe−iθ)
= [1/2i] (e2iθ − e0 + e0 − e−2iθ)
= [(e2iθ − e−2iθ)/2i]

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.