Solving 4 Tan² Θ = 4 Tan Θ Find Θ In The Interval -π To Π
Introduction: Navigating Trigonometric Equations
In the realm of mathematics, trigonometric equations often present intriguing challenges, requiring a blend of algebraic manipulation and a solid grasp of trigonometric identities. This article delves into the solution of the equation 4 tan² θ = 4 tan θ, focusing on identifying all values of θ that satisfy this equation within the specified interval of -π ≤ θ < π. We will embark on a step-by-step journey, employing algebraic techniques to simplify the equation and trigonometric principles to pinpoint the desired solutions. This exploration will not only enhance your problem-solving skills but also deepen your understanding of the behavior of trigonometric functions.
Deconstructing the Equation: A Step-by-Step Approach
Our initial task is to isolate the variable and unravel the equation 4 tan² θ = 4 tan θ. To achieve this, we employ algebraic manipulation, aiming to bring all terms to one side and factorize the expression. This strategic move allows us to identify potential solutions by setting each factor equal to zero. First, let's subtract 4 tan θ from both sides of the equation:
4 tan² θ - 4 tan θ = 0
Now, we observe a common factor of 4 tan θ on the left-hand side. Factoring this out, we get:
4 tan θ (tan θ - 1) = 0
This factorization is a pivotal step, transforming the original equation into a product of two factors. According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. Thus, we have two possibilities:
- 4 tan θ = 0
- tan θ - 1 = 0
Let's tackle each possibility individually to determine the values of θ that satisfy the equation. For the first case, 4 tan θ = 0, we divide both sides by 4, leading to tan θ = 0. This implies that θ must be an angle whose tangent is zero. Recall that the tangent function represents the ratio of sine to cosine, so tan θ = 0 when sin θ = 0. Within the interval -π ≤ θ < π, the angles that satisfy this condition are θ = -π, 0. Let's move on to the second case, tan θ - 1 = 0. Adding 1 to both sides, we get tan θ = 1. This means we are looking for angles whose tangent is equal to 1. The tangent function is positive in the first and third quadrants. Within our interval, the angle in the first quadrant whose tangent is 1 is θ = π/4, and the angle in the third quadrant is θ = -3π/4. By systematically breaking down the equation and considering the properties of the tangent function, we have identified four potential solutions for θ. In the next section, we will consolidate these solutions and present the complete answer.
Identifying Solutions: Zero Tangent and Unit Tangent
Unveiling Angles with Zero Tangent
As established in the previous section, the equation tan θ = 0 holds significance in our quest for solutions. The tangent function, defined as the ratio of sine to cosine (tan θ = sin θ / cos θ), vanishes when the sine function is zero. Consequently, we seek angles θ within the interval -π ≤ θ < π where sin θ = 0. Recall the unit circle, where the sine function corresponds to the y-coordinate of a point on the circle. The y-coordinate is zero at the points where the circle intersects the x-axis. These intersections occur at multiples of π. Within our specified interval, we find two angles where sin θ = 0: θ = 0 and θ = -π. At θ = 0, the point on the unit circle is (1, 0), and at θ = -π, the point is (-1, 0). In both instances, the y-coordinate, representing the sine value, is zero. Therefore, θ = 0 and θ = -π are solutions to the equation tan θ = 0 within the given interval. These solutions are crucial components of the overall solution set for the original equation, 4 tan² θ = 4 tan θ. Understanding the relationship between the tangent function and the sine function is essential for solving trigonometric equations effectively. By recognizing that tan θ = 0 when sin θ = 0, we were able to identify the angles θ = 0 and θ = -π as key solutions. Now, let's explore the second scenario, where tan θ = 1, to complete our solution-finding journey.
Discovering Angles with Unit Tangent
Having explored the angles where the tangent function is zero, we now turn our attention to the second critical case: tan θ = 1. This condition seeks angles θ within the interval -π ≤ θ < π where the tangent function assumes a value of 1. Recall that tan θ = sin θ / cos θ, so tan θ = 1 when sin θ = cos θ. Geometrically, this signifies that the y-coordinate and x-coordinate of the point on the unit circle corresponding to θ are equal. This occurs in the first and third quadrants. In the first quadrant, the angle whose sine and cosine are equal is θ = π/4. At this angle, both sin(π/4) and cos(π/4) are equal to √2/2, making tan(π/4) = 1. In the third quadrant, both sine and cosine are negative, but their ratio remains positive. The angle in the third quadrant that satisfies tan θ = 1 is θ = -3π/4. At this angle, sin(-3π/4) and cos(-3π/4) are both equal to -√2/2, resulting in tan(-3π/4) = 1. Therefore, θ = π/4 and θ = -3π/4 are solutions to the equation tan θ = 1 within the specified interval. These solutions, along with the previously found solutions for tan θ = 0, collectively form the complete solution set for the original equation. By understanding the unit circle and the relationship between sine, cosine, and tangent, we were able to identify these solutions effectively. In the next section, we will synthesize these individual solutions into a comprehensive answer.
Synthesizing the Solutions: The Complete Picture
Having meticulously dissected the equation 4 tan² θ = 4 tan θ and explored the scenarios where tan θ = 0 and tan θ = 1, we now stand ready to assemble the complete solution set within the interval -π ≤ θ < π. Our journey began with algebraic manipulation, leading to the factored form 4 tan θ (tan θ - 1) = 0. This pivotal step unveiled two distinct possibilities: tan θ = 0 and tan θ = 1. For tan θ = 0, we delved into the unit circle and the relationship between tangent and sine, identifying the angles θ = 0 and θ = -π as solutions. These angles correspond to the points on the unit circle where the y-coordinate, representing the sine value, is zero. Subsequently, we tackled the case of tan θ = 1, seeking angles where the sine and cosine functions are equal. This led us to the solutions θ = π/4 in the first quadrant and θ = -3π/4 in the third quadrant. At these angles, the ratio of sine to cosine is unity, satisfying the condition tan θ = 1. Now, we consolidate these individual solutions into a comprehensive set. The values of θ that satisfy the equation 4 tan² θ = 4 tan θ within the interval -π ≤ θ < π are θ = -π, 0, π/4, and -3π/4. This complete solution set represents all the angles within the given interval that make the original equation true. By systematically breaking down the problem, exploring the properties of the tangent function, and utilizing the unit circle as a visual aid, we have successfully navigated this trigonometric equation. The solutions we have found are not merely numerical answers; they are geometric representations of angles that satisfy a specific trigonometric relationship. In the final section, we will formally present our answer, highlighting the key steps and insights gained throughout this exploration.
Conclusion: The Answer Unveiled
In conclusion, the values of θ that satisfy the equation 4 tan² θ = 4 tan θ within the interval -π ≤ θ < π are θ = -π, 0, π/4, -3π/4. This solution was achieved through a systematic approach, beginning with algebraic manipulation to factor the equation, followed by a detailed exploration of the conditions tan θ = 0 and tan θ = 1. The unit circle served as a valuable tool in visualizing the relationships between sine, cosine, and tangent, enabling us to identify the angles that meet these conditions. This exploration underscores the importance of a strong foundation in trigonometric identities and algebraic techniques for solving trigonometric equations. By combining these skills, we can effectively navigate complex problems and arrive at accurate solutions. The process of solving this equation has not only provided us with a numerical answer but also deepened our understanding of the behavior of trigonometric functions and their geometric interpretations. We have seen how the tangent function, as the ratio of sine to cosine, plays a crucial role in determining the solutions. The angles -π, 0, π/4, and -3π/4 represent specific points on the unit circle where the tangent function exhibits the desired behavior, either vanishing or assuming a value of 1. This journey through trigonometric problem-solving has reinforced the power of a methodical approach and the beauty of mathematical relationships. The solutions we have found are a testament to the interconnectedness of algebra, trigonometry, and geometry, highlighting the elegance and precision of mathematics.
