Solving Trigonometric Equations Find X Between 0 And 90 Degrees
Trigonometric equations, especially those involving sine and cosine, often present intriguing challenges. This article delves into the methods for solving such equations, specifically focusing on problems where we need to find the value of 'x' within the range of 0° to 90°. We'll explore the fundamental trigonometric identities and properties that allow us to simplify and solve these equations systematically. Our discussion will center around two key problems that exemplify common scenarios encountered in trigonometry. We'll also emphasize the importance of understanding the relationships between trigonometric functions and how they can be manipulated to arrive at solutions.
Understanding the Problem
This problem requires us to find the value of x that satisfies the equation cos(3x + 14) = sin(x - 20), given the constraint 0 ≤ x ≤ 90°. The key to solving this lies in understanding the relationship between sine and cosine functions. We know that sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ). These identities are crucial for transforming the equation into a solvable form. By applying these identities, we can express both sides of the equation in terms of either sine or cosine, allowing us to equate the angles and solve for x. This method is particularly useful when dealing with equations that mix sine and cosine functions. We'll also need to be mindful of the given range for x to ensure that our solution falls within the specified interval. Furthermore, understanding the periodicity and symmetry of sine and cosine functions can help us identify potential multiple solutions, but in this case, the constraint on x simplifies the process by limiting the solution to a single value.
Solution
To solve the equation cos(3x + 14) = sin(x - 20), we can use the identity cos(θ) = sin(90° - θ). Applying this to the left side of the equation, we get:
sin(90° - (3x + 14)) = sin(x - 20)
Simplifying the angle on the left side:
sin(90° - 3x - 14) = sin(x - 20)
sin(76° - 3x) = sin(x - 20)
Now that both sides of the equation are in terms of sine, we can equate the angles:
76° - 3x = x - 20°
Adding 3x to both sides and adding 20° to both sides gives:
76° + 20° = x + 3x
96° = 4x
Dividing both sides by 4, we find:
x = 24°
Since 24° falls within the range 0 ≤ x ≤ 90°, this is a valid solution.
Verification
To ensure our solution is correct, we can substitute x = 24° back into the original equation:
cos(3(24) + 14) = cos(72 + 14) = cos(86°)
sin(24 - 20) = sin(4°)
Using the identity cos(θ) = sin(90° - θ), we have:
cos(86°) = sin(90° - 86°) = sin(4°)
Thus, the solution x = 24° is verified.
Understanding the Problem
This second problem, cos(4x - 28) = sin(2x - 50), follows a similar structure to the first but introduces different angles within the trigonometric functions. Again, we are tasked with finding the value of x within the range of 0° to 90°. The core strategy remains the same: we will leverage the complementary relationship between sine and cosine. By converting either the cosine to sine or vice versa, we can create an equation where both sides involve the same trigonometric function. This allows us to equate the angles and form a linear equation in terms of x. The algebraic manipulation required to isolate x will be a crucial step, and we must meticulously handle the coefficients and constants. Finally, once we obtain a potential value for x, we must check if it lies within the specified range of 0° to 90° to confirm its validity. Understanding and applying trigonometric identities is paramount to successfully navigating this problem.
Solution
Similar to the first problem, we use the identity cos(θ) = sin(90° - θ) to rewrite the equation cos(4x - 28) = sin(2x - 50).
Applying the identity to the left side, we get:
sin(90° - (4x - 28)) = sin(2x - 50)
Simplifying the angle:
sin(90° - 4x + 28) = sin(2x - 50)
sin(118° - 4x) = sin(2x - 50)
Equating the angles:
118° - 4x = 2x - 50°
Adding 4x to both sides and adding 50° to both sides gives:
118° + 50° = 2x + 4x
168° = 6x
Dividing both sides by 6, we find:
x = 28°
Since 28° falls within the range 0 ≤ x ≤ 90°, this is a valid solution.
Verification
To verify the solution, we substitute x = 28° back into the original equation:
cos(4(28) - 28) = cos(112 - 28) = cos(84°)
sin(2(28) - 50) = sin(56 - 50) = sin(6°)
Using the identity cos(θ) = sin(90° - θ), we have:
cos(84°) = sin(90° - 84°) = sin(6°)
Thus, the solution x = 28° is verified.
In summary, solving trigonometric equations such as cos(3x + 14) = sin(x - 20) and cos(4x - 28) = sin(2x - 50), within a specified range, requires a strong understanding of trigonometric identities and algebraic manipulation. The key is to use identities like cos(θ) = sin(90° - θ) to transform the equation into a form where both sides involve the same trigonometric function. This allows for the angles to be equated, leading to a linear equation that can be easily solved for x. Verification of the solution by substituting it back into the original equation is a crucial step to ensure accuracy. The problems discussed here exemplify a common type of trigonometric problem, and the methods used can be applied to a wide range of similar equations. Mastering these techniques is essential for anyone studying trigonometry and related fields. Furthermore, the emphasis on understanding the underlying relationships between trigonometric functions provides a solid foundation for tackling more complex problems in the future. Remember, practice is key to proficiency in mathematics, and consistent application of these methods will build confidence and skill.
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