Physics 171 Math Review - Free Printable

Let’s go step by step and solve each problem carefully.

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1. Express x in terms of a, b and c
This is the quadratic equation:
ax² + bx + c = 0

We use the quadratic formula:
x = [-b ± √(b² - 4ac)] / (2a)

✔ Final Answer for #1:
x = (-b ± √(b² - 4ac)) / (2a)

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2. Logarithm rules

- ln(eⁿ) = n → because natural log and e cancel out
- ln(x·y) = ln(x) + ln(y) → product rule
- ln(x/y) = ln(x) - ln(y) → quotient rule
- ln(aˣ) = x·ln(a) → power rule
- e ≈ 2.718 → to 3 digits

✔ Final Answers for #2:
ln(eⁿ) = n
ln(x·y) = ln(x) + ln(y)
ln(x/y) = ln(x) - ln(y)
ln(aˣ) = x·ln(a)
e = 2.718

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3. Common logarithms (base 10)

- log(10ⁿ) = n → same idea as above, but base 10
- log(1000) = log(10³) = 3
- log(0.01) = log(10⁻²) = -2

✔ Final Answers for #3:
log(10ⁿ) = n
log(1000) = 3
log(0.01) = -2

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4. Circle formulas

- Area of circle = πr²
- Circumference of circle = 2πr
- 360° = 2π radians → full circle
- π ≈ 3.14 → to 3 digits

✔ Final Answers for #4:
Area of circle = πr²
Circumference of circle = 2πr
360° = 2π radians
π = 3.14

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5. Right triangle with sides a, b, c (c = hypotenuse), angles α and β

From diagram:
- Side opposite α is a
- Side adjacent to α is b
- Hypotenuse is c
- Angle β is at top, so side opposite β is b, adjacent is a

Express c in terms of a and b → Pythagorean theorem:
c = √(a² + b²)

Express α in terms of β → since it’s a right triangle, α + β = 90° → α = 90° - β

Area of triangle = (1/2) × base × height = (1/2)ab

✔ Final Answers for #5:
c = √(a² + b²)
α = 90° - β
Area of triangle = (1/2)ab

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6. Trig ratios in right triangle

Using standard definitions:

sin α = opposite/hypotenuse = a/c
cos α = adjacent/hypotenuse = b/c
sin β = opposite/hypotenuse = b/c
cos β = adjacent/hypotenuse = a/c
tan α = opposite/adjacent = a/b

✔ Final Answers for #6:
sin α = a/c
cos α = b/c
sin β = b/c
cos β = a/c
tan α = a/b

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7. Trig identities using angle α

Use symmetry and co-function identities:

sin(-α) = -sin α → sine is odd
cos(-α) = cos α → cosine is even
sin(90° - α) = cos α
cos(90° - α) = sin α
sin(180° - α) = sin α → second quadrant, sine positive
sin(180° + α) = -sin α → third quadrant, sine negative

✔ Final Answers for #7:
sin(-α) = -sin α
cos(-α) = cos α
sin(90° - α) = cos α
cos(90° - α) = sin α
sin(180° - α) = sin α
sin(180° + α) = -sin α

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8. Exact trig values

sin 0° = 0
cos 0° = 1
sin 90° = 1
cos 90° = 0
sin 180° = 0
cos 180° = -1
sin 270° = -1
cos 270° = 0
tan 0° = 0
tan 45° = 1

✔ Final Answers for #8:
sin 0° = 0
cos 0° = 1
sin 90° = 1
cos 90° = 0
sin 180° = 0
cos 180° = -1
sin 270° = -1
cos 270° = 0
tan 0° = 0
tan 45° = 1

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9. Law of Sines or Cosines? Diagram shows general triangle with sides A, B, C and angles α, β, γ

Actually, looking at labels:
Side A is opposite angle α
Side B is opposite angle β
Side C is opposite angle γ

So we can use Law of Cosines to express one side in terms of others and included angle.

To express C in terms of A, B, and γ:
C² = A² + B² - 2AB·cos(γ) → so C = √(A² + B² - 2AB·cos(γ))

Similarly:
B² = A² + C² - 2AC·cos(β) → B = √(A² + C² - 2AC·cos(β))
A² = B² + C² - 2BC·cos(α) → A = √(B² + C² - 2BC·cos(α))

✔ Final Answers for #9:
C = √(A² + B² - 2AB·cos γ)
B = √(A² + C² - 2AC·cos β)
A = √(B² + C² - 2BC·cos α)

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Final Answer:
1. x = (-b ± √(b² - 4ac)) / (2a)
2. ln(eⁿ) = n; ln(x·y) = ln(x)+ln(y); ln(x/y)=ln(x)-ln(y); ln(aˣ)=x·ln(a); e=2.718
3. log(10ⁿ)=n; log(1000)=3; log(0.01)=-2
4. Area=πr²; Circumference=2πr; 360°=2π rad; π=3.14
5. c=√(a²+b²); α=90°−β; Area=(1/2)ab
6. sinα=a/c; cosα=b/c; sinβ=b/c; cosβ=a/c; tanα=a/b
7. sin(-α)=-sinα; cos(-α)=cosα; sin(90°−α)=cosα; cos(90°−α)=sinα; sin(180°−α)=sinα; sin(180°+α)=-sinα
8. sin0°=0; cos0°=1; sin90°=1; cos90°=0; sin180°=0; cos180°=-1; sin270°=-1; cos270°=0; tan0°=0; tan45°=1
9. C=√(A²+B²−2AB·cosγ); B=√(A²+C²−2AC·cosβ); A=√(B²+C²−2BC·cosα)