What is the method for solving cosine squared equations?

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    Cosine Method
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SUMMARY

The method for solving the equation cos²(θ) = 0.6 involves using trigonometric identities and manual calculations. To solve by hand, one can take the square root of 0.6 and apply the identity cos(2θ) = 2cos²(θ) - 1, leading to the equation 2cos²(θ) - 1 = 0.2. This results in θ being approximately π/4 - 0.1, which can be verified using a calculator to confirm that cos²(π/4 - 0.1) ≈ 0.5993.

PREREQUISITES
  • Understanding of trigonometric identities, specifically cos(2θ) = 2cos²(θ) - 1
  • Ability to perform square root calculations manually
  • Familiarity with basic trigonometric functions and their values
  • Knowledge of using cosine tables for angle approximations
NEXT STEPS
  • Study the derivation and applications of the double angle formulas in trigonometry
  • Practice solving trigonometric equations without a calculator
  • Learn how to use cosine tables effectively for angle calculations
  • Explore numerical methods for solving trigonometric equations
USEFUL FOR

Students in mathematics or physics, educators teaching trigonometry, and anyone interested in manual problem-solving techniques for trigonometric equations.

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I was just finishing my physics homework (don't worry this isn't a HW question, the homework is done) and the last calculation I had to do was cos2(θ) = 0.6. I just plugged this into my calculator to solve for me and got it right.

Now I'm curious though. How would one solve this by hand?
 
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HunterDX77M said:
I was just finishing my physics homework (don't worry this isn't a HW question, the homework is done) and the last calculation I had to do was cos2(θ) = 0.6. I just plugged this into my calculator to solve for me and got it right.

Now I'm curious though. How would one solve this by hand?

Take the square root of 0.6 by hand, and use cosine tables?
 
HunterDX77M said:
cos2(θ) = 0.6. I just plugged this into my calculator to solve for me and got it right.
Now I'm curious though. How would one solve this by hand?
If you really mean 'by hand' (no calculator),
cos(2θ) = 2 cos2(θ) -1 = 0.2 = sin(π/2 - 2θ)
π/2 - 2θ ≈ 0.2
θ ≈ π/4 - 0.1
Check (by calculator): cos2(π/4 - 0.1) = 0.5993..
 

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