Finding Cos(2x) When Sin(x) And Cos(x) Are Given A Trigonometric Solution

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Understanding trigonometric functions and their relationships is crucial in mathematics. This article delves into a specific problem involving the sine and cosine functions, providing a step-by-step solution and explanation. Let's explore how to find the value of cos(2x) when given the value of sin(x) and a condition on cos(x).

Problem Statement

Suppose $\sin (x)=-\frac{3}{5}$ and $\cos (x) < 0$. What is the value of $\cos (2 x)$?

A. $\frac{-16}{25}$ B. $\frac{-7}{25}$ C. $\frac{7}{25}$ D. $\frac{16}{25}$

Understanding the Problem: Sine, Cosine, and Double Angle Formula

To solve this problem effectively, it's essential to grasp the fundamental concepts of trigonometric functions, specifically sine and cosine, and their relationships. The sine function, denoted as sin(x), and the cosine function, denoted as cos(x), are fundamental in trigonometry and describe the ratios of sides in a right-angled triangle, further extended to the unit circle. In this context, sin(x) represents the y-coordinate and cos(x) represents the x-coordinate of a point on the unit circle corresponding to the angle x. The unit circle, with a radius of 1, provides a visual and mathematical framework for understanding trigonometric functions for all real numbers, not just angles within a triangle. The signs of sin(x) and cos(x) in different quadrants of the unit circle are crucial: in the first quadrant (0 to 90 degrees), both are positive; in the second (90 to 180 degrees), sin(x) is positive and cos(x) is negative; in the third (180 to 270 degrees), both are negative; and in the fourth (270 to 360 degrees), cos(x) is positive and sin(x) is negative. Additionally, the Pythagorean identity, sin²(x) + cos²(x) = 1, is a cornerstone in trigonometry, linking sine and cosine and allowing us to find one if the other is known. This identity arises directly from the Pythagorean theorem applied to the unit circle. The double-angle formula for cosine, cos(2x) = cos²(x) - sin²(x), is another critical tool, enabling us to express the cosine of double an angle in terms of the sine and cosine of the original angle. There are alternative forms of this formula, such as cos(2x) = 2cos²(x) - 1 and cos(2x) = 1 - 2sin²(x), which can be derived using the Pythagorean identity. Choosing the appropriate form can simplify calculations depending on the given information. For example, if sin(x) is known, using cos(2x) = 1 - 2sin²(x) might be more straightforward. These foundational concepts and identities form the basis for solving a wide range of trigonometric problems, including the one presented in this article.

Step-by-Step Solution

1. Determine cos(x) using the Pythagorean Identity.

We are given that $\sin(x) = -\frac{3}{5}$. We also know the fundamental Pythagorean identity:

sin⁡2(x)+cos⁡2(x)=1\sin^2(x) + \cos^2(x) = 1

Substitute the value of sin(x) into the equation:

(−35)2+cos⁡2(x)=1\left(-\frac{3}{5}\right)^2 + \cos^2(x) = 1

925+cos⁡2(x)=1\frac{9}{25} + \cos^2(x) = 1

cos⁡2(x)=1−925\cos^2(x) = 1 - \frac{9}{25}

cos⁡2(x)=1625\cos^2(x) = \frac{16}{25}

Taking the square root of both sides gives us two possible values for cos(x):

cos⁡(x)=±45\cos(x) = \pm \frac{4}{5}

2. Choose the correct value of cos(x) based on the given condition.

We are given that $\cos(x) < 0$. This means that cos(x) is negative. Therefore, we choose the negative value:

cos⁡(x)=−45\cos(x) = -\frac{4}{5}

3. Apply the double-angle formula for cosine.

We need to find the value of $\cos(2x)$. There are several double-angle formulas for cosine, but the most convenient one to use here is:

cos⁡(2x)=cos⁡2(x)−sin⁡2(x)\cos(2x) = \cos^2(x) - \sin^2(x)

Substitute the values of sin(x) and cos(x) into the formula:

cos⁡(2x)=(−45)2−(−35)2\cos(2x) = \left(-\frac{4}{5}\right)^2 - \left(-\frac{3}{5}\right)^2

cos⁡(2x)=1625−925\cos(2x) = \frac{16}{25} - \frac{9}{25}

cos⁡(2x)=725\cos(2x) = \frac{7}{25}

4. Final Answer

Therefore, the value of $\cos(2x)$ is $\frac{7}{25}$. So the correct answer is C. This methodical approach, combining trigonometric identities with the given conditions, ensures an accurate solution. The use of the Pythagorean identity and the double-angle formula for cosine are key steps in solving this type of problem. Understanding these concepts allows for the application of similar techniques in a variety of trigonometric challenges. The crucial step of considering the sign of cos(x) based on the given condition highlights the importance of a thorough analysis of the problem statement. This ensures that the selected value of cos(x) is consistent with the information provided, leading to the correct final answer. In essence, the solution demonstrates the power of combining algebraic manipulation with a solid understanding of trigonometric principles. This combination is fundamental in tackling more complex trigonometric problems and in various applications of trigonometry in physics, engineering, and other scientific disciplines.

Alternative Approach using another Double-Angle Formula

An alternative approach to solving this problem involves using a different form of the double-angle formula for cosine. Instead of using $,cos(2x) = cos^2(x) - sin^2(x)$, we can use the formula that directly involves $,sin(x)$: $,cos(2x) = 1 - 2sin^2(x)$. This approach can be more straightforward in some cases, especially when the value of $,sin(x)$ is already known. Let's walk through the steps using this alternative formula.

1. Recall the given information.

We are given that $,sin(x) = -\frac{3}{5}$. Our goal is to find the value of $,cos(2x)$.

2. Apply the alternative double-angle formula.

The formula we will use is:

 cos(2x)=1−2sin2(x)\,cos(2x) = 1 - 2sin^2(x)

3. Substitute the value of sin(x).

Substitute the given value of $,sin(x)$ into the formula:

 cos(2x)=1−2(−35)2\,cos(2x) = 1 - 2\left(-\frac{3}{5}\right)^2

4. Simplify the expression.

First, square the value of $,sin(x)$:

(−35)2=925\left(-\frac{3}{5}\right)^2 = \frac{9}{25}

Now, substitute this back into the formula:

 cos(2x)=1−2(925)\,cos(2x) = 1 - 2\left(\frac{9}{25}\right)

Multiply 2 by $,\frac{9}{25}$:

 cos(2x)=1−1825\,cos(2x) = 1 - \frac{18}{25}

5. Find a common denominator and subtract.

To subtract $,\frac{18}{25}$ from 1, we need to express 1 as a fraction with a denominator of 25:

1=25251 = \frac{25}{25}

Now, subtract:

 cos(2x)=2525−1825\,cos(2x) = \frac{25}{25} - \frac{18}{25}

 cos(2x)=725\,cos(2x) = \frac{7}{25}

6. Final Answer.

Using this alternative method, we arrive at the same answer: $,cos(2x) = \frac{7}{25}$. This confirms the result we obtained using the first method. This alternative approach demonstrates the flexibility in using trigonometric identities to solve problems. By choosing the formula that best suits the given information, we can often simplify the calculations and arrive at the solution more efficiently. In this case, since the value of $,sin(x)$ was readily available, using the formula $,cos(2x) = 1 - 2sin^2(x)$ allowed us to bypass the step of finding $,cos(x)$ explicitly. This is a valuable strategy in problem-solving, as it highlights the importance of recognizing and utilizing the available information effectively. Both methods, however, underscore the fundamental principles of trigonometry and the consistent relationships between trigonometric functions. The ability to apply these principles flexibly is a key skill in trigonometry and mathematics in general.

Key Takeaways

  • Understanding the Pythagorean identity and double-angle formulas is crucial for solving trigonometric problems.
  • Always consider the given conditions (e.g., the sign of cos(x)) to determine the correct solution.
  • There can be multiple approaches to solving a trigonometric problem; choose the one that seems most efficient for the given information.
  • Double-check your work to ensure accuracy, especially when dealing with signs and fractions.

By mastering these concepts and techniques, you can confidently tackle a wide range of trigonometric problems. Remember to practice and apply these principles to various scenarios to strengthen your understanding and problem-solving skills. This article has provided a detailed explanation of how to find the value of cos(2x) given specific conditions for sin(x) and cos(x). The step-by-step solution, along with the alternative approach, illustrates the importance of understanding trigonometric identities and their applications. By consistently applying these principles, you can enhance your proficiency in trigonometry and tackle more complex problems with confidence. The ability to manipulate trigonometric identities and choose the most appropriate formula for a given situation is a valuable skill in mathematics and its applications in science and engineering. Furthermore, the emphasis on considering the sign of trigonometric functions based on the given conditions is a crucial aspect of problem-solving in trigonometry. This ensures that the solution is not only mathematically correct but also consistent with the specific context of the problem. In essence, the key to success in trigonometry lies in a solid understanding of the fundamental identities, the ability to apply them flexibly, and a meticulous approach to problem analysis and solution verification.

Conclusion

In conclusion, solving trigonometric problems like this requires a solid understanding of trigonometric identities and careful consideration of given conditions. By applying the Pythagorean identity and the double-angle formula, we were able to determine the value of $,cos(2x)$. The ability to choose the appropriate formula and method is a key skill in trigonometry. The example presented in this article highlights the interconnectedness of trigonometric concepts and the importance of a methodical approach to problem-solving. By understanding the relationships between sine, cosine, and other trigonometric functions, and by being able to manipulate these functions using identities, one can successfully navigate a wide range of trigonometric challenges. The alternative method presented further underscores the versatility of trigonometric identities and the importance of choosing the most efficient approach for a given problem. The key takeaway is that a strong foundation in trigonometric principles, coupled with practice and careful attention to detail, is essential for success in this area of mathematics. Trigonometry is not merely a collection of formulas and equations; it is a framework for understanding and modeling periodic phenomena, with applications ranging from physics and engineering to music and art. Therefore, mastering the concepts and techniques presented in this article is a valuable step towards a deeper appreciation of mathematics and its role in the world around us. The problem-solving strategies discussed here are not only applicable to trigonometric problems but also to a broader range of mathematical and scientific challenges. The ability to break down a complex problem into smaller, manageable steps, to identify relevant information and constraints, and to apply appropriate tools and techniques is a hallmark of effective problem-solving in any field.