Math Problem Statement

give me the solution of this math question

Solution

Let's re-examine the problem and solve it step-by-step.

Given the parametric equations: x(t)=cos(t)+cos(12t)x(t) = \cos(t) + \cos\left(\frac{1}{2} t\right) y(t)=sin(t)+sin(12t)y(t) = \sin(t) + \sin\left(\frac{1}{2} t\right)

The point PP passes through the x-axis at the origin and three more times. This occurs when y(t)=0y(t) = 0.

Step 1: Set y(t)y(t) to 0

sin(t)+sin(12t)=0\sin(t) + \sin\left(\frac{1}{2} t\right) = 0

Step 2: Solve for tt

Using the sum-to-product identities: sin(t)+sin(12t)=2sin(t+12t2)cos(t12t2)\sin(t) + \sin\left(\frac{1}{2} t\right) = 2 \sin\left(\frac{t + \frac{1}{2} t}{2}\right) \cos\left(\frac{t - \frac{1}{2} t}{2}\right) =2sin(34t)cos(14t)=0= 2 \sin\left(\frac{3}{4} t\right) \cos\left(\frac{1}{4} t\right) = 0

This product equals zero if either factor is zero:

  1. sin(34t)=0\sin\left(\frac{3}{4} t\right) = 0
  2. cos(14t)=0\cos\left(\frac{1}{4} t\right) = 0

Step 3: Solve sin(34t)=0\sin\left(\frac{3}{4} t\right) = 0

34t=nπ\frac{3}{4} t = n \pi t=43nπt = \frac{4}{3} n \pi

Where nn is any integer. We need to find values of tt within the given range 2πt2π-2\pi \leq t \leq 2\pi.

Possible values of tt:

  • t=0t = 0
  • t=43πt = \frac{4}{3} \pi
  • t=83πt = \frac{8}{3} \pi
  • t=43πt = -\frac{4}{3} \pi
  • t=83πt = -\frac{8}{3} \pi

Step 4: Solve cos(14t)=0\cos\left(\frac{1}{4} t\right) = 0

14t=(n+12)π\frac{1}{4} t = \left(n + \frac{1}{2}\right) \pi t=4(n+12)πt = 4\left(n + \frac{1}{2}\right) \pi t=(4n+2)πt = (4n + 2) \pi

Where nn is any integer. We need to find values of tt within the given range 2πt2π-2\pi \leq t \leq 2\pi.

Possible values of tt:

  • t=2πt = 2\pi
  • t=2πt = -2\pi

Step 5: Combine and filter the solutions

We discard t=0t = 0 since we are looking for values where PP crosses the x-axis besides the origin. We also discard ±2π\pm 2\pi as they are outside the given range.

Thus, the valid values of tt are: t1=43πt_1 = \frac{4}{3} \pi t2=43πt_2 = -\frac{4}{3} \pi t3=83πt_3 = \frac{8}{3} \pi

Final Answer

[ [t_1, t_2, t_3] = \left[ \frac{4}{3} \pi, -\frac{4}{3} \pi, \frac{8}{3} \pi \right] ]

If you have any further questions or need more details, feel free to ask! Here are some questions you might be interested in:

  1. How can I verify the solutions for tt?
  2. What is the geometric interpretation of this parametric equation?
  3. How do I apply these parametric equations in a different context?
  4. Can you provide a graph for better visualization?
  5. Are there alternative methods to solve this problem?

Tip: When solving trigonometric equations, always consider using identities to simplify the expressions before solving them.

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Math Problem Analysis

Mathematical Concepts

Parametric Equations
Trigonometric Identities

Formulas

Sum-to-product identities

Theorems

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Suitable Grade Level

Advanced High School