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asambeis [7]
3 years ago
9

Find the values of θ at which there are horizontal tangent lines on the graph of r = 1 + cos θ.

Mathematics
1 answer:
Amiraneli [1.4K]3 years ago
7 0

Answer:

θ = π/3 and 5π/3

Step-by-step explanation:

x = r cos θ and y = r sin θ.  Therefore:

dy/dx = (dy/dθ) / (dx/dθ)

dy/dx = (r cos θ + r' sin θ) / (-r sin θ + r' cos θ)

In this case, r = 1 + cos θ and r' = -sin θ.

dy/dx = ((1 + cos θ) cos θ + (-sin θ) sin θ) / (-(1 + cos θ) sin θ + (-sin θ) cos θ)

dy/dx = (cos θ + cos²θ − sin²θ) / (-sin θ − sin θ cos θ − sin θ cos θ)

dy/dx = (cos θ + cos²θ − (1 − cos²θ)) / (-sin θ (1 + 2 cos θ))

dy/dx = (cos θ + 2 cos²θ − 1) / (-sin θ (1 + 2 cos θ))

The tangent line is horizontal when the numerator of dy/dx is 0 and the denominator of dy/dx is not 0.

0 = 2 cos²θ + cos θ − 1

0 = (cos θ + 1) (2 cos θ − 1)

cos θ = -1 or 1/2

θ = π/3, π, 5π/3

0 = -sin θ (1 + 2 cos θ)

sin θ = 0 or cos θ = -1/2

θ = 0, 2π/3, π, 4π/3

Therefore, the answer is θ = π/3 and 5π/3.

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Answer:

51,111%

Step-by-step explanation:

We know that 45% have a dog but no cat.  While 23% have both a dog and a cat.

Then Pr(X has cat|X has dog) = Pr(X has dog and cat)/Pr(X has dog)

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6 0
3 years ago
What is the diameter of the circle?
Hatshy [7]
Hello!

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I hope this helps!
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3 years ago
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What is the missing value in the proportion 4/5 = x/30?
ra1l [238]

Answer:

\displaystyle x = 24

General Formulas and Concepts:

<u>Pre-Algebra</u>

Order of Operations: BPEMDAS

  1. Brackets
  2. Parenthesis
  3. Exponents
  4. Multiplication
  5. Division
  6. Addition
  7. Subtraction
  • Left to Right  

Equality Properties

  • Multiplication Property of Equality
  • Division Property of Equality
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Step-by-step explanation:

<u>Step 1: Define</u>

<u />\displaystyle \frac{4}{5} = \frac{x}{30}<u />

<u />

<u>Step 2: Solve for </u><em><u>x</u></em>

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3 years ago
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Delicious77 [7]
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3 years ago
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