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Solve 8cos(5x)=3 for the smallest three positive solutions. Give your answers (in radians) accurate to at least two decimal places, as a list separated by commas.

Solve 8cos⁡(5x)=3 8 \cos (5 x)=3 for the smallest three positive solutions.\newlineGive your answers (in radians) accurate to at least two decimal places, as a list separated by commas.

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Full solution

Q. Solve 8cos⁡(5x)=3 8 \cos (5 x)=3 for the smallest three positive solutions.\newlineGive your answers (in radians) accurate to at least two decimal places, as a list separated by commas.
  1. Isolate Cosine Function: Isolate the cosine function by dividing both sides of the equation by 88.\newline8cos⁡(5x)=3 8\cos(5x) = 3 \newlinecos⁡(5x)=38 \cos(5x) = \frac{3}{8}
  2. Find General Solution: Find the general solution for x by taking the arccosine of both sides.\newline5x=arccos⁡(38) 5x = \arccos\left(\frac{3}{8}\right) \newlinex=15arccos⁡(38) x = \frac{1}{5}\arccos\left(\frac{3}{8}\right)
  3. Calculate Principal Value: Calculate the arccosine value to find the principal value for x.\newlinex=15arccos⁡(38)≈15×1.230959 x = \frac{1}{5}\arccos\left(\frac{3}{8}\right) \approx \frac{1}{5} \times 1.230959 \newlinex≈0.246192 x \approx 0.246192 \newlineThis is the first positive solution.
  4. Determine Second Solution: Determine the second solution using the periodicity of the cosine function.\newlineCosine is periodic with a period of 2π2\pi, so the next solution will be:\newlinex=15arccos⁡(38)+2π5 x = \frac{1}{5}\arccos\left(\frac{3}{8}\right) + \frac{2\pi}{5} \newlinex≈0.246192+2π5 x \approx 0.246192 + \frac{2\pi}{5} \newlinex≈0.246192+1.256637 x \approx 0.246192 + 1.256637 \newlinex≈1.502829 x \approx 1.502829 \newlineThis is the second positive solution.
  5. Find Third Solution: Find the third solution by adding another period of 2π2\pi to the second solution.\newlinex≈1.502829+2π5 x \approx 1.502829 + \frac{2\pi}{5} \newlinex≈1.502829+1.256637 x \approx 1.502829 + 1.256637 \newlinex≈2.759466 x \approx 2.759466 \newlineThis is the third positive solution.

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