How do you solve for cos2x = 2 cosx sinx?

  • Context: High School 
  • Thread starter Thread starter Maria
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SUMMARY

The equation cos(2x) = 2cos(x)sin(x) can be transformed using the identity 2cos(x)sin(x) = sin(2x), simplifying the problem to finding the angles where cos(2x) = sin(2x). This leads to the equation tan(2x) = 1, which results in four solutions for x within the range of 0 to 360 degrees: 22.5°, 112.5°, 202.5°, and 292.5°. The solutions are derived by setting 2x = 45 + 180n, where n is an integer, ensuring all angles are accounted for.

PREREQUISITES
  • Understanding of trigonometric identities, specifically sin(2x) and cos(2x)
  • Knowledge of the tangent function and its periodicity
  • Ability to solve equations involving angles and periodic functions
  • Familiarity with the unit circle and angle measurement in degrees
NEXT STEPS
  • Study the derivation and applications of trigonometric identities such as sin(2x) and cos(2x)
  • Learn how to solve trigonometric equations involving multiple angles, focusing on periodic functions
  • Explore the implications of solving tan(kx) = 1 for various integer values of k
  • Practice finding all solutions within specified ranges for trigonometric equations
USEFUL FOR

Students studying trigonometry, educators teaching trigonometric identities, and anyone looking to deepen their understanding of solving trigonometric equations.

  • #31
I have on stupid question left:
Why do I get 4 angles instead og just 2?
Is it because tan =1?
 
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  • #32
You get 4 distinct angles because you must solve tan(2x)=1 rather than tan(x)=1
If for example, you were to solve tan(4x)=1, you would have even more distinct solutions
(You could work out how many for yourself)
 
  • #33
thanks a lot for answering all my stupid questions...
 

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