How do you solve #\cos 2x = - 0.9#?

Answer 1

Use the inverse cosine. #x=1/2cos^-1(-0.9) +-pin#

To isolate #x#, we can start by taking the inverse cosine of both sides, given by #cos^-1# or #arccos#. You can think of the inverse cosine undoing the cosine function like taking the natural log of #e^x# gets us back to #x# if it helps.
#cos(f(x))=a=>f(x)=arccos(a) color(grey)(+2pin)#
Given #cos(2x)=-0.9#:
#2x=cos^-1(-0.9)+2pin#
Now divide by #2#:
#x=1/2cos^-1(-0.9) +-pin#
We use the period to find the full set of all solutions. The period of #cos(x)# is #2pi#, so these solutions repeat every #2pi# units.
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Answer 2

To solve the equation (\cos 2x = -0.9), follow these steps:

  1. Use the double-angle identity for cosine: (\cos 2x = 2\cos^2 x - 1).
  2. Substitute (-0.9) for (\cos 2x): (2\cos^2 x - 1 = -0.9).
  3. Rearrange the equation: (2\cos^2 x = -0.9 + 1).
  4. Simplify: (2\cos^2 x = 0.1).
  5. Divide both sides by 2: (\cos^2 x = 0.05).
  6. Take the square root of both sides: (\cos x = \pm \sqrt{0.05}).
  7. Calculate the square root of (0.05).
  8. Since cosine can be both positive and negative, we have two cases: a. (\cos x = \sqrt{0.05}) b. (\cos x = -\sqrt{0.05}).
  9. Use the inverse cosine function to find the values of (x): a. For (\cos x = \sqrt{0.05}), (x = \arccos(\sqrt{0.05}) + 2\pi n) and (x = -\arccos(\sqrt{0.05}) + 2\pi n). b. For (\cos x = -\sqrt{0.05}), (x = \arccos(-\sqrt{0.05}) + 2\pi n) and (x = -\arccos(-\sqrt{0.05}) + 2\pi n).
  10. Calculate the numerical values of (x) using a calculator or computer software.
  11. Include (2\pi n) to account for all possible solutions, where (n) is an integer.
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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