Finding Cos(t) Given Sin(t) = 3/5 And T In The Second Quadrant

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In trigonometry, determining the values of trigonometric functions like cosine, sine, tangent, etc., is a common task. Often, we are given the value of one trigonometric function and the quadrant in which the angle lies, and we need to find the values of the other functions. This article focuses on finding the value of $\cos(t)$\ given $\sin(t)$\ and the quadrant of t. Specifically, we will address the problem where $\sin(t) = \frac{3}{5}$\ and t is in the second quadrant. Understanding these types of problems is fundamental for various applications in mathematics, physics, and engineering. Mastering trigonometric identities and quadrant rules allows for precise and efficient problem-solving. The following sections will detail the step-by-step process to solve this particular problem and offer insights into the underlying principles and techniques involved.

Understanding the Problem

Given Information

We are given that $\sin(t) = \frac{3}{5}$\ and that t lies in the second quadrant. This information is crucial because the quadrant determines the sign of the trigonometric functions. In the second quadrant, sine is positive, cosine is negative, tangent is negative, cosecant is positive, secant is negative, and cotangent is negative. Knowing $\sin(t)$\ and the quadrant, we can find $\cos(t)$\ using trigonometric identities and quadrant rules. Trigonometric identities provide relationships between different trigonometric functions, allowing us to express one in terms of another. For instance, the Pythagorean identity is a cornerstone in solving such problems. Additionally, understanding the unit circle and how trigonometric functions behave in different quadrants is essential. The sign of $\cos(t)$\ will be negative in the second quadrant, which is a critical piece of information for arriving at the correct solution. The approach involves using the Pythagorean identity to relate sine and cosine, followed by considering the quadrant to determine the correct sign for cosine.

Strategy

To find $\cos(t)$\, we will use the Pythagorean identity:

sin⁡2(t)+cos2(t)=1\\\sin^2(t) + \\cos^2(t) = 1

This identity relates sine and cosine, allowing us to find $\cos(t)$\ if we know $\sin(t)$\. First, we will substitute the given value of $\sin(t)$\ into the identity. Then, we will solve for $\cos^2(t)$\ and subsequently for $\cos(t)$\. Since t is in the second quadrant, we know that $\cos(t)$\ is negative. Therefore, we will choose the negative square root when solving for $\cos(t)$\. This step is crucial because the square root operation yields both positive and negative solutions, but the quadrant information helps us select the correct sign. By following this strategy, we can systematically determine the value of $\cos(t)$\ while ensuring the solution aligns with the given conditions. This method highlights the importance of not only trigonometric identities but also the context provided by quadrant information.

Step-by-Step Solution

Step 1: Using the Pythagorean Identity

The Pythagorean identity states that:

sin⁡2(t)+cos2(t)=1\\\sin^2(t) + \\cos^2(t) = 1

We are given $\sin(t) = \frac{3}{5}$\, so we substitute this into the identity:

left(frac35right)2+cos2(t)=1\\left(\\frac{3}{5}\\right)^2 + \\cos^2(t) = 1

Squaring $\frac{3}{5}$\ gives us:

925+cos2(t)=1\\\frac{9}{25} + \\cos^2(t) = 1

Step 2: Solving for $\cos^2(t)$\

To isolate $\cos^2(t)$\, we subtract $\frac{9}{25}$\ from both sides of the equation:

cos2(t)=1−frac925\\cos^2(t) = 1 - \\frac{9}{25}

To subtract the fractions, we need a common denominator, which is 25. So, we rewrite 1 as $\frac{25}{25}$\:

cos2(t)=frac2525−frac925\\cos^2(t) = \\frac{25}{25} - \\frac{9}{25}

cos2(t)=frac1625\\cos^2(t) = \\frac{16}{25}

Step 3: Solving for $\cos(t)$\

Now, we take the square root of both sides to solve for $\cos(t)$\:

cos(t)=pmsqrtfrac1625\\cos(t) = \\pm\\sqrt{\\frac{16}{25}}

cos(t)=pmfrac45\\cos(t) = \\pm\\frac{4}{5}

Step 4: Determining the Sign of $\cos(t)$\

Since t is in the second quadrant, cosine is negative. In the second quadrant, the x-coordinates are negative, which corresponds to the cosine values. Therefore, we choose the negative root:

cos(t)=−frac45\\cos(t) = -\\frac{4}{5}

Final Answer

Given that $\sin(t) = \frac{3}{5}$\ and t is in the second quadrant, we have found that:

cos(t)=−frac45\\cos(t) = -\\frac{4}{5}

This result aligns with the properties of trigonometric functions in the second quadrant, where sine is positive and cosine is negative. The step-by-step solution demonstrates the application of the Pythagorean identity and the importance of considering quadrant information to determine the correct sign of the trigonometric functions. Understanding these principles is crucial for solving more complex trigonometric problems. The process involves using trigonometric identities to relate different functions, algebraic manipulation to isolate the desired function, and applying quadrant rules to ascertain the correct sign. This comprehensive approach ensures accurate and reliable solutions in trigonometry.

Conclusion

In summary, we have successfully found the value of $\cos(t)$\ given $\sin(t) = \frac{3}{5}$\ and the information that t is in the second quadrant. The key steps involved using the Pythagorean identity $\\sin^2(t) + \cos^2(t) = 1$\, solving for $\cos^2(t)$\, taking the square root to find $\cos(t)$\, and then determining the correct sign based on the quadrant. In the second quadrant, cosine is negative, so we chose the negative root. This problem illustrates the importance of understanding trigonometric identities and how the signs of trigonometric functions vary across different quadrants. Mastering these concepts is essential for solving a wide range of trigonometric problems and applications in various fields, including physics, engineering, and mathematics. This exercise also highlights the significance of a systematic approach to problem-solving, which includes identifying the given information, selecting the appropriate trigonometric identities, performing algebraic manipulations, and interpreting the results in the context of the given conditions. By following these steps, we can accurately and efficiently solve trigonometric problems.