Calculating Cos(Ɵ) Given Sin(Ɵ) Equals 2/3 A Step By Step Guide
In the realm of trigonometry, understanding the relationships between trigonometric functions is crucial. One common problem involves finding the cosine of an angle (cos(Ɵ)) when the sine of the same angle (sin(Ɵ)) is known. This article delves into a step-by-step approach to solve this problem, focusing on the scenario where sin(Ɵ) = 2/3. We will explore the fundamental trigonometric identities, demonstrate the calculation process, and discuss the implications of the solutions obtained. This comprehensive guide aims to provide a clear and detailed explanation, making it accessible for students and enthusiasts alike.
Understanding the Fundamentals
Before diving into the calculation, it's essential to grasp the fundamental trigonometric identities that govern the relationship between sine, cosine, and other trigonometric functions. The cornerstone of this calculation is the Pythagorean identity, which states that sin²(Ɵ) + cos²(Ɵ) = 1. This identity is derived from the Pythagorean theorem applied to the unit circle, where the radius is 1, and the coordinates of a point on the circle are (cos(Ɵ), sin(Ɵ)). The Pythagorean identity serves as the bridge connecting sine and cosine, allowing us to determine one if the other is known.
The Pythagorean Identity: sin²(Ɵ) + cos²(Ɵ) = 1
The Pythagorean identity is a direct consequence of the Pythagorean theorem (a² + b² = c²) applied to a right-angled triangle inscribed in a unit circle. In this context, sin(Ɵ) represents the y-coordinate, cos(Ɵ) represents the x-coordinate, and the radius of the unit circle is 1. Squaring the sine and cosine values and adding them together gives us the square of the radius, which is 1. This identity is fundamental in trigonometry and is used extensively in various applications, including solving trigonometric equations, simplifying expressions, and analyzing wave phenomena.
The significance of the Pythagorean identity lies in its ability to relate sine and cosine directly. Given the value of one function, we can use this identity to find the value of the other, up to a sign. For example, if we know sin(Ɵ), we can rearrange the identity to solve for cos(Ɵ), and vice versa. This is particularly useful in problems where only one trigonometric function is given, and the others need to be determined.
Moreover, the Pythagorean identity is not limited to acute angles; it holds true for all angles, making it a universally applicable tool in trigonometry. Whether the angle is in the first quadrant or the fourth quadrant, the relationship between sin²(Ɵ) and cos²(Ɵ) remains consistent. This broad applicability makes the identity indispensable in trigonometric analysis and problem-solving.
In the context of our problem, where sin(Ɵ) = 2/3, the Pythagorean identity provides the direct pathway to calculating cos(Ɵ). By substituting the given value of sin(Ɵ) into the identity, we can isolate cos²(Ɵ) and subsequently find cos(Ɵ). This process highlights the power and elegance of the identity in simplifying trigonometric calculations.
Other Important Trigonometric Identities
While the Pythagorean identity is the primary tool for solving our problem, it's beneficial to be aware of other trigonometric identities. These identities can be useful in more complex scenarios and provide a deeper understanding of trigonometric relationships. Some key identities include:
- Tangent Identity: tan(Ɵ) = sin(Ɵ) / cos(Ɵ)
- Cotangent Identity: cot(Ɵ) = cos(Ɵ) / sin(Ɵ)
- Secant Identity: sec(Ɵ) = 1 / cos(Ɵ)
- Cosecant Identity: csc(Ɵ) = 1 / sin(Ɵ)
- **Angle Sum and Difference Identities:
- sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B)
- cos(A ± B) = cos(A)cos(B) ∓ sin(A)sin(B)
- **Double Angle Identities:
- sin(2Ɵ) = 2sin(Ɵ)cos(Ɵ)
- cos(2Ɵ) = cos²(Ɵ) - sin²(Ɵ) = 1 - 2sin²(Ɵ) = 2cos²(Ɵ) - 1
These identities, along with the Pythagorean identity, form the foundation of trigonometric manipulation and problem-solving. Understanding these relationships allows for the simplification of complex expressions, the solution of trigonometric equations, and the analysis of periodic functions.
Step-by-Step Calculation of cos(Ɵ) when sin(Ɵ) = 2/3
Now, let's apply the Pythagorean identity to calculate cos(Ɵ) when sin(Ɵ) is given as 2/3. This process involves a straightforward substitution and algebraic manipulation to isolate cos(Ɵ).
Step 1: Substitute the Given Value into the Pythagorean Identity
We begin with the Pythagorean identity: sin²(Ɵ) + cos²(Ɵ) = 1. We are given that sin(Ɵ) = 2/3. Substituting this value into the identity, we get:
(2/3)² + cos²(Ɵ) = 1
This substitution replaces the sin²(Ɵ) term with its numerical equivalent, allowing us to proceed with solving for cos²(Ɵ).
Step 2: Simplify the Equation
Next, we simplify the equation by squaring the fraction (2/3)²:
4/9 + cos²(Ɵ) = 1
This step simplifies the equation and prepares it for isolating cos²(Ɵ).
Step 3: Isolate cos²(Ɵ)
To isolate cos²(Ɵ), we subtract 4/9 from both sides of the equation:
cos²(Ɵ) = 1 - 4/9
This manipulation moves the constant term to the right side of the equation, leaving cos²(Ɵ) by itself on the left side.
Step 4: Calculate cos²(Ɵ)
Now, we perform the subtraction:
cos²(Ɵ) = 9/9 - 4/9
cos²(Ɵ) = 5/9
This step yields the value of cos²(Ɵ).
Step 5: Find cos(Ɵ) by Taking the Square Root
To find cos(Ɵ), we take the square root of both sides of the equation:
cos(Ɵ) = ±√(5/9)
This step is crucial as it introduces the possibility of both positive and negative solutions for cos(Ɵ).
Step 6: Simplify the Square Root
We simplify the square root:
cos(Ɵ) = ±√5 / √9
cos(Ɵ) = ±√5 / 3
This simplification provides the final solutions for cos(Ɵ).
Interpreting the Results: Two Possible Values for cos(Ɵ)
The calculation yields two possible values for cos(Ɵ): √5 / 3 and -√5 / 3. This result highlights a critical aspect of trigonometric functions: for a given value of sin(Ɵ), there are generally two possible angles within the range of 0 to 2π (or 0° to 360°) that satisfy the condition. This is because the sine function is positive in both the first and second quadrants, while the cosine function can be positive or negative depending on the quadrant.
The Significance of the Plus/Minus Sign
The presence of both positive and negative solutions for cos(Ɵ) stems from the nature of the cosine function in different quadrants of the unit circle. In the first quadrant (0° to 90°), both sine and cosine are positive. In the second quadrant (90° to 180°), sine is positive, but cosine is negative. This means that if sin(Ɵ) = 2/3, there is an angle in the first quadrant where cos(Ɵ) is positive and an angle in the second quadrant where cos(Ɵ) is negative.
To determine the correct value of cos(Ɵ), additional information is needed, such as the quadrant in which the angle Ɵ lies. If the angle is specified to be in the first quadrant, then cos(Ɵ) = √5 / 3. If the angle is in the second quadrant, then cos(Ɵ) = -√5 / 3.
Visualizing the Solutions on the Unit Circle
Understanding the unit circle is crucial for visualizing trigonometric functions and their values. The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. Angles are measured counterclockwise from the positive x-axis. The x-coordinate of a point on the unit circle corresponds to the cosine of the angle, and the y-coordinate corresponds to the sine of the angle.
In our case, sin(Ɵ) = 2/3 means that the y-coordinate of the point on the unit circle is 2/3. There are two points on the unit circle with this y-coordinate: one in the first quadrant and one in the second quadrant. The x-coordinate of the point in the first quadrant is √5 / 3, corresponding to cos(Ɵ) = √5 / 3. The x-coordinate of the point in the second quadrant is -√5 / 3, corresponding to cos(Ɵ) = -√5 / 3.
This visualization reinforces the concept that there are two possible angles for a given sine value, each with a different cosine value. To pinpoint the correct solution, the quadrant of the angle must be known.
Conclusion
Calculating cos(Ɵ) when sin(Ɵ) = 2/3 demonstrates the application of the Pythagorean identity and the importance of understanding the signs of trigonometric functions in different quadrants. The two possible solutions, √5 / 3 and -√5 / 3, underscore the need for additional information to determine the specific angle being considered. By mastering these concepts, one can confidently tackle a wide range of trigonometric problems and gain a deeper appreciation for the relationships between trigonometric functions. This comprehensive guide has provided a detailed explanation of the process, making it accessible for students and enthusiasts alike to further explore the fascinating world of trigonometry.
By understanding the Pythagorean identity and its implications, we can solve trigonometric problems effectively. The step-by-step approach outlined in this article provides a clear methodology for calculating cos(Ɵ) when sin(Ɵ) is known. Remember, the plus/minus sign in the solution highlights the two possible quadrants where the angle can lie, emphasizing the importance of additional information to determine the specific value of cos(Ɵ). Trigonometry is not just about calculations; it's about understanding the underlying relationships and visualizing the concepts, which is key to mastering this branch of mathematics.