Find The Exact Value Of Each X[0,2) For Which Sin(2x)= 3cos(x)
Understanding how to solve trigonometric equations such as Sin(2x) = 3Cos(x) within a specific interval like [0, 2) is fundamental in mathematics, especially in calculus, algebra, and trigonometry. This article provides a comprehensive guide to finding the exact values of all solutions for x in the interval [0, 2), where the equation Sin(2x) = 3Cos(x) holds true. We will explore the problem step-by-step, including relevant identities, solution methods, and detailed explanations to ensure clarity and mastery.
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Understanding the Trigonometric Equation
Before diving into the solution, it’s essential to understand the components of the given equation:
- Sin(2x): The sine of double the angle x.
- Cos(x): The cosine of angle x.
- The equation equates these two functions scaled by a factor of 3, i.e., Sin(2x) = 3Cos(x).
The goal is to find all x within the interval [0, 2), where this equality holds exactly.
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Key Trigonometric Identities and Concepts
To approach this problem effectively, familiarize yourself with the following identities:
Double-Angle Identity for Sine
- Sin(2x) = 2Sin(x)Cos(x)
Basic Trigonometric Range
- Cos(x) and Sin(x) have ranges [-1, 1], which restricts possible solutions because the right side involves a multiple of Cos(x).
Interval [0, 2)
- The solutions sought are within x ∈ [0, 2) (radians), covering from 0 up to but not including 2 radians.
Step-by-Step Solution Process
The overall plan involves transforming the original equation into a solvable form and then analyzing the solutions within the given interval.
Step 1: Rewrite the Equation Using Identities
Starting from:
\[ \sin(2x) = 3 \cos(x) \]
Apply the double-angle identity:
\[ 2 \sin(x) \cos(x) = 3 \cos(x) \]
Step 2: Simplify the Equation
Divide both sides by cos(x), but be cautious about values where cos(x) = 0:
\[ 2 \sin(x) = 3 \]
or
\[ \text{if } \cos(x) \neq 0 \quad \Rightarrow \quad \sin(x) = \frac{3}{2} \]
Since \(\sin(x)\) can only be in [-1, 1], \(\sin(x) = \frac{3}{2}\) is impossible in the real domain. This indicates that solutions where cos(x) ≠ 0 do not exist for this equation.
Step 3: Consider When \(\cos(x) = 0\)
The division by cos(x) is invalid where cos(x) = 0. Let's analyze these cases separately.
- Set \(\cos(x) = 0\):
\[ \cos(x) = 0 \]
- The solutions for x in [0, 2):
\[ x = \frac{\pi}{2} \]
since:
\[ \cos\left(\frac{\pi}{2}\right) = 0 \]
- Now, check if x = \(\frac{\pi}{2}\) satisfies the original equation:
\[ \sin(2 \times \frac{\pi}{2}) = 3 \cos\left(\frac{\pi}{2}\right) \]
Calculate:
\[ \sin(\pi) = 3 \times 0 \]
\[ 0 = 0 \]
This is true, so x = \(\frac{\pi}{2}\) is a solution.
- Summary: When cos(x) = 0, x = \(\pi/2\) is a solution.
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Summary of Solutions
From the above analysis:
- When cos(x) ≠ 0, the only possible solution would require \(\sin(x) = 3/2\), which is impossible in real numbers.
- When cos(x) = 0, the only solution in the interval [0, 2) is x = \(\pi/2\), which satisfies the original equation.
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Final Solution Set within [0, 2)
Based on the analysis, the only solution in the interval [0, 2) is:
- x = \(\pi/2\)
since:
\[ \pi/2 \approx 1.5708 \]
which lies within [0, 2).
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Verification of the Solution
Always verify solutions by substituting back into the original equation:
- At x = \(\pi/2\):
\[ \sin(2 \times \pi/2) = \sin(\pi) = 0 \]
\[ 3 \cos(\pi/2) = 3 \times 0 = 0 \]
Both sides are equal (0), confirming x = \(\pi/2\) is a valid solution.
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Additional Considerations and Domain Restrictions
- Since the initial step involved division by cos(x), solutions where cos(x) = 0 needed separate analysis, which we did and found valid solutions.
- No other solutions exist because of the range restriction of the sine function, \(\sin(x) \leq 1\), and the fact that \(\sin(x) = 3/2\) is outside the permissible range.
- Thus, the only exact solution in [0, 2) is x = \(\pi/2\).
Conclusion
In conclusion, the only solution to the equation Sin(2x) = 3Cos(x) within the interval [0, 2) is:
\[ \boxed{x = \frac{\pi}{2}} \]
This solution satisfies all the necessary conditions and has been verified explicitly. The analysis highlights the importance of considering special cases where denominators could be zero and understanding the ranges of trigonometric functions.
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Additional Tips for Solving Similar Trigonometric Equations
- Always check for solutions when dividing by trig functions to avoid missing solutions where the divisor is zero.
- Use fundamental identities to rewrite complex equations in simpler forms.
- Remember the ranges of sine and cosine functions to eliminate impossible solutions.
- Verify solutions by substitution back into the original equation.
- Be mindful of the interval domain when solving equations in restricted domains.
Meta Description:
Learn how to find the exact value of x in the interval [0, 2) for the equation Sin(2x) = 3Cos(x). This comprehensive guide explores solution methods, identities, and verification techniques for mastering trigonometric equations.
