Solving For Sin(2θ) Using Trigonometric Identities And Quadrant Information

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In trigonometry, solving problems often requires a deep understanding of trigonometric identities and the relationships between angles in different quadrants. This article delves into a specific problem: given that cos(θ) = -3√10/10 and θ is in Quadrant III, our goal is to determine the value of sin(2θ). This exploration will not only provide a step-by-step solution to this problem but also reinforce the fundamental concepts that underpin trigonometric problem-solving.

Determining sin(θ) using the Pythagorean Identity and Quadrant Information

The problem provides us with the value of cos(θ) and the information that θ lies in Quadrant III. To find sin(2θ), we first need to determine the value of sin(θ). This is where the Pythagorean identity comes into play. The Pythagorean identity, a cornerstone of trigonometry, states that sin²(θ) + cos²(θ) = 1. This identity allows us to relate the sine and cosine of an angle, enabling us to find one if we know the other.

In our case, we know cos(θ) = -3√10/10. Substituting this value into the Pythagorean identity, we get:

sin²(θ) + (-3√10/10)² = 1

Simplifying the equation:

sin²(θ) + (9 * 10) / 100 = 1 sin²(θ) + 90/100 = 1 sin²(θ) + 9/10 = 1

Now, we isolate sin²(θ) by subtracting 9/10 from both sides:

sin²(θ) = 1 - 9/10 sin²(θ) = 1/10

To find sin(θ), we take the square root of both sides:

sin(θ) = ±√(1/10) sin(θ) = ±1/√10

At this point, we have two possible values for sin(θ): 1/√10 and -1/√10. This is where the quadrant information becomes crucial. We are given that θ is in Quadrant III. In Quadrant III, both sine and cosine are negative. Therefore, we choose the negative value for sin(θ):

sin(θ) = -1/√10

To rationalize the denominator, we multiply the numerator and denominator by √10:

sin(θ) = -√10/10

So, we have successfully determined that sin(θ) = -√10/10. This step was crucial as it provides the necessary building block for calculating sin(2θ).

Applying the Double Angle Identity for Sine

Now that we have the values of both sin(θ) and cos(θ), we can move on to the next step: finding sin(2θ). This requires the use of the double angle identity for sine. The double angle identity for sine states that:

sin(2θ) = 2sin(θ)cos(θ)

This identity is a direct consequence of the angle addition formula for sine and is a fundamental tool in trigonometric manipulations. It allows us to express the sine of twice an angle in terms of the sine and cosine of the angle itself. Now, we can substitute the values we have already determined: sin(θ) = -√10/10 and cos(θ) = -3√10/10 into this identity:

sin(2θ) = 2 * (-√10/10) * (-3√10/10)

Now, let's simplify the expression step by step:

sin(2θ) = 2 * (3 * 10) / 100 sin(2θ) = 60 / 100

Finally, we reduce the fraction to its simplest form:

sin(2θ) = 3/5

Therefore, the value of sin(2θ) is 3/5. This is the solution to the problem.

Summary and Key Takeaways

In this exploration, we successfully found the value of sin(2θ) given cos(θ) = -3√10/10 and the fact that θ is in Quadrant III. The process involved several key steps, each highlighting important trigonometric concepts:

  1. Using the Pythagorean Identity: We used the Pythagorean identity (sin²(θ) + cos²(θ) = 1) to find sin(θ) from cos(θ).
  2. Quadrant Information: We used the information about the quadrant of θ to determine the correct sign of sin(θ).
  3. Double Angle Identity: We applied the double angle identity (sin(2θ) = 2sin(θ)cos(θ)) to calculate sin(2θ).

This problem demonstrates the interconnectedness of trigonometric concepts and the importance of understanding trigonometric identities and quadrant relationships. By mastering these concepts, one can confidently tackle a wide range of trigonometric problems.

Understanding the relationships between trigonometric functions and their values in different quadrants is crucial for solving problems like this one. The Pythagorean identity and double-angle formulas are powerful tools in trigonometry, and this example showcases their practical application. By breaking down the problem into smaller steps and carefully applying these concepts, we arrived at the solution: sin(2θ) = 3/5. This exercise not only provides a specific answer but also strengthens our understanding of trigonometric principles.

In conclusion, solving for sin(2θ) in this scenario required a multi-faceted approach, combining the Pythagorean identity, quadrant analysis, and the double angle formula. The detailed walkthrough presented here serves as a valuable resource for anyone seeking to enhance their trigonometry skills and problem-solving abilities. Remember, consistent practice and a solid grasp of fundamental principles are key to success in mathematics.

Trigonometry, double angle identity, Pythagorean identity, quadrant, sine, cosine, sin(2θ), cos(θ), sin(θ), unit circle, angle, solution, mathematics