Math Problem Statement
Solution
To solve the equation , we can proceed by expressing the equation in a single trigonometric form.
Steps to Solve
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Combine and into one term: We use the concept of expressing as a single cosine or sine function.
Recall the identity: where and is an angle satisfying and .
Here, and , so:
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Determine and :
Therefore:
Using the cosine addition formula , we rewrite:
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Substitute into the original equation: The equation becomes:
Solving for , divide both sides by :
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Solve for : The general solution for is:
Therefore:
Substituting back for , where , we can determine :
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Write the final solution for : Substituting , the solution simplifies to:
Final Notes:
The solution involves two angles:
- , which is ,
- .
You can compute approximate numerical values for these angles using a calculator or software.
Let me know if you need further clarification or a step-by-step numerical approximation! 😊
Related Questions:
- How do you solve trigonometric equations using identities?
- What is the general solution of ?
- How do you express as a single cosine function?
- How do you determine if ?
- What are the applications of trigonometric equations in real life?
Tip: Expressing sums like as a single trigonometric function simplifies solving equations efficiently.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Trigonometric Identities
Cosine and Sine Functions
Formulas
aCos(x) + bSin(x) = R Cos(x - φ), where R = √(a² + b²)
Cos(A - B) = Cos(A)Cos(B) + Sin(A)Sin(B)
General solution for Cos(θ) = c: θ = ± arccos(c) + 2nπ
Theorems
Trigonometric addition formulas
Pythagorean theorem
Suitable Grade Level
Grades 11-12