Finding Cos Θ Given Tan Θ In Quadrant I A Trigonometry Problem
Delving into the fascinating world of trigonometry, we encounter a multitude of functions and relationships that govern the behavior of angles and their corresponding sides in triangles. Among these, the tangent (tan), cosine (cos), and sine (sin) functions hold paramount importance. These functions play a crucial role in various fields, including physics, engineering, and navigation. In this article, we embark on a journey to unravel the intricacies of trigonometric functions, focusing on a specific problem that involves finding the value of cos θ given the value of tan θ and the quadrant in which θ lies. Mastering these concepts will empower you to tackle a wide range of trigonometric problems and gain a deeper appreciation for the elegance and power of mathematics.
Before we dive into the specific problem at hand, let's take a moment to refresh our understanding of the fundamental trigonometric functions: sine (sin), cosine (cos), and tangent (tan). These functions relate the angles of a right-angled triangle to the ratios of its sides. Consider a right-angled triangle with an angle θ, where:
- The opposite side is the side opposite to angle θ.
- The adjacent side is the side adjacent to angle θ.
- The hypotenuse is the side opposite the right angle.
Then, the trigonometric functions are defined as follows:
- Sine (sin θ) = Opposite / Hypotenuse
- Cosine (cos θ) = Adjacent / Hypotenuse
- Tangent (tan θ) = Opposite / Adjacent
These functions are not limited to right-angled triangles alone. They can be extended to any angle using the unit circle, a circle with a radius of 1 centered at the origin of a coordinate plane. The coordinates of a point on the unit circle corresponding to an angle θ are given by (cos θ, sin θ). This representation allows us to define trigonometric functions for angles beyond the range of 0 to 90 degrees.
Quadrants and Trigonometric Signs
The coordinate plane is divided into four quadrants, numbered I to IV, starting from the top right quadrant and proceeding counterclockwise. The signs of trigonometric functions vary depending on the quadrant in which the angle lies. This is due to the signs of the x and y coordinates in each quadrant.
- Quadrant I (0° < θ < 90°): All trigonometric functions (sin, cos, tan) are positive.
- Quadrant II (90° < θ < 180°): Sine (sin) is positive, while cosine (cos) and tangent (tan) are negative.
- Quadrant III (180° < θ < 270°): Tangent (tan) is positive, while sine (sin) and cosine (cos) are negative.
- Quadrant IV (270° < θ < 360°): Cosine (cos) is positive, while sine (sin) and tangent (tan) are negative.
Understanding the signs of trigonometric functions in different quadrants is crucial for solving trigonometric problems, as it helps us determine the correct sign of the answer.
Now, let's turn our attention to the specific problem we aim to solve. We are given the following equation:
tan(θ) = 4 / 2
We are also told that θ is an angle in quadrant I. Our task is to find the value of cos(θ). This problem combines our knowledge of trigonometric functions and their behavior in different quadrants.
To solve this problem, we will employ a step-by-step approach, leveraging our understanding of trigonometric identities and the relationship between tangent, sine, and cosine.
Step 1: Simplify the Tangent Value
The given equation is:
tan(θ) = 4 / 2
Simplifying the fraction, we get:
tan(θ) = 2
This simplified value of tan(θ) will be easier to work with in subsequent steps.
Step 2: Use the Pythagorean Identity
Recall the fundamental Pythagorean identity:
1 + tan²(θ) = sec²(θ)
where sec(θ) is the secant function, which is the reciprocal of cosine:
sec(θ) = 1 / cos(θ)
We can use this identity to find sec²(θ) and then cos²(θ).
Substitute the value of tan(θ) into the Pythagorean identity:
1 + (2)² = sec²(θ)
1 + 4 = sec²(θ)
sec²(θ) = 5
Step 3: Find Cosine Squared
Now that we have sec²(θ), we can find cos²(θ) using the reciprocal relationship:
cos²(θ) = 1 / sec²(θ)
cos²(θ) = 1 / 5
Step 4: Find Cosine
To find cos(θ), we take the square root of both sides:
cos(θ) = ±√(1 / 5)
cos(θ) = ±(1 / √5)
Step 5: Determine the Sign
Since θ is in quadrant I, where cosine is positive, we choose the positive value:
cos(θ) = 1 / √5
Step 6: Rationalize the Denominator (Optional)
To rationalize the denominator, we multiply the numerator and denominator by √5:
cos(θ) = (1 / √5) * (√5 / √5)
cos(θ) = √5 / 5
Therefore, the value of cos(θ) is:
cos(θ) = √5 / 5
This completes our solution to the problem. We have successfully found the value of cos(θ) given the value of tan(θ) and the quadrant in which θ lies. This problem demonstrates the interconnectedness of trigonometric functions and the importance of understanding their properties and relationships.
In this article, we have explored the realm of trigonometric functions, focusing on a problem that involved finding the value of cos θ given the value of tan θ and the quadrant in which θ lies. We began by revisiting the fundamental definitions of sine, cosine, and tangent, and then delved into the concept of quadrants and the signs of trigonometric functions in each quadrant. We then tackled the specific problem, employing a step-by-step approach that involved simplifying the tangent value, using the Pythagorean identity, finding cosine squared, determining the sign of cosine, and rationalizing the denominator. Through this process, we gained a deeper understanding of the relationships between trigonometric functions and the importance of considering the quadrant in which the angle lies.
Mastering trigonometric functions is essential for success in various fields, including mathematics, physics, engineering, and navigation. By understanding the definitions, properties, and relationships of these functions, we can solve a wide range of problems and gain a deeper appreciation for the elegance and power of mathematics. As you continue your journey in mathematics, remember to practice regularly and seek out challenging problems to further enhance your understanding and skills. The world of trigonometry awaits your exploration!
