Need help solving this trig equation

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SUMMARY

The discussion focuses on solving the trigonometric equation 3cos(θ) + 1.595*sin(θ) = 3.114. The initial approach using the Pythagorean identity (sinθ)^2 + (cosθ)^2 = 1 led to an incorrect value for cosθ, specifically +/- 1.0526, which is outside the valid range for cosine. A more effective method involves transforming the equation into the form a cos(θ) + b sin(θ) = R cos(θ - φ), where R = √(a² + b²) and φ = arctan(b/a), allowing for a correct solution for θ.

PREREQUISITES
  • Understanding of trigonometric identities, specifically the Pythagorean identity.
  • Familiarity with the transformation of trigonometric equations into the form a cos(θ) + b sin(θ).
  • Knowledge of the range of trigonometric functions, particularly the cosine function.
  • Basic understanding of right triangle properties and the tangent function.
NEXT STEPS
  • Study the derivation of the transformation a cos(θ) + b sin(θ) = R cos(θ - φ).
  • Practice solving various trigonometric equations using the Pythagorean identity.
  • Explore the implications of the range of trigonometric functions on potential solutions.
  • Learn about the graphical representation of trigonometric functions to visualize solutions.
USEFUL FOR

Students studying trigonometry, educators teaching trigonometric equations, and anyone looking to improve their problem-solving skills in trigonometric contexts.

flgdx
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Homework Statement


what's the best way to solve this equation: 3cos(θ) + 1.595*sin(θ) = 3.114

Homework Equations


(sinθ)^2 + (cosθ)^2 = 1

The Attempt at a Solution


I tried using the identity above to solve this equation and ended up with cosθ = +/- 1.0526.
 
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Could you show your steps for your solution?
 
Comeback City said:
Could you show your steps for your solution?

so since (sinθ)^2 + (cosθ)^2 = 1, I solved for sinθ and got sqrt(1 - (cosθ)^2). Then I subtituted it into the equation 3cos(θ) + 1.595*sin(θ) = 3.114 and solved for θ.
 
flgdx said:

Homework Statement


what's the best way to solve this equation: 3cos(θ) + 1.595*sin(θ) = 3.114

Homework Equations


(sinθ)^2 + (cosθ)^2 = 1

The Attempt at a Solution


I tried using the identity above to solve this equation and ended up with cosθ = +/- 1.0526.
Well, that's obviously wrong because -1 ≤ cos θ ≤ 1, and 1.0526 is outside that range. Your approach should work though.

Another approach is to use
$$a \cos\theta + b \sin\theta = \sqrt{a^2+b^2} \cos(\theta-\phi)$$ where ##\tan\phi = b/a##.
 
could you pls show me how you got that equation?
 
Consider a right triangle below and use the identity ##\cos(\theta-\phi) = \cos\theta\,\cos\phi + \sin\theta\,\sin\phi##.

triangle.png
 

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