Evaluate Sine And Tangent Given Cosine And Quadrant

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In the realm of trigonometry, understanding the relationships between trigonometric functions and their values within specific quadrants is crucial. This article delves into a problem where we are given the cosine of an angle, cos⁡(θ)=22\cos(\theta) = \frac{\sqrt{2}}{2}, and the quadrant in which the angle lies, 3π2<θ<2π\frac{3\pi}{2} < \theta < 2\pi. Our mission is to evaluate the sine, sin⁡(θ)\sin(\theta), and tangent, tan⁡(θ)\tan(\theta), of this angle. This exploration will not only reinforce fundamental trigonometric identities but also highlight the significance of quadrant analysis in determining the signs of trigonometric functions.

Decoding the Given Information

Understanding the Cosine Value

We are given that cos⁡(θ)=22\cos(\theta) = \frac{\sqrt{2}}{2}. The cosine function, in the unit circle context, represents the x-coordinate of a point on the circle corresponding to the angle θ\theta. A cosine value of 22\frac{\sqrt{2}}{2} is associated with angles that have a reference angle of π4\frac{\pi}{4} (45 degrees). This is because cos⁡(π4)=22\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}. However, since cosine is positive in both the first and fourth quadrants, we need additional information to pinpoint the exact location of θ\theta.

Quadrant Specification: 3π/2 < θ < 2π

The inequality 3π2<θ<2π\frac{3\pi}{2} < \theta < 2\pi tells us that θ\theta lies in the fourth quadrant. This is a crucial piece of information. In the fourth quadrant:

  • Cosine is positive.
  • Sine is negative.
  • Tangent is negative.

This quadrant restriction confirms that our given cosine value is consistent with the location of the angle. It also dictates the sign of the sine and tangent values we are about to calculate. This quadrant is where the magic happens, as it narrows down the possibilities and guides us to the correct solution. Ignoring this crucial piece of information can lead to incorrect results. Therefore, always pay close attention to the given quadrant when solving trigonometric problems. This understanding is fundamental to correctly evaluating trigonometric functions, especially when dealing with angles beyond the first revolution.

Evaluating Sine (sin(θ))

To find sin⁡(θ)\sin(\theta), we can employ the fundamental Pythagorean trigonometric identity:

sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1

We know cos⁡(θ)=22\cos(\theta) = \frac{\sqrt{2}}{2}, so we can substitute this value into the identity:

sin⁡2(θ)+(22)2=1\sin^2(\theta) + (\frac{\sqrt{2}}{2})^2 = 1

sin⁡2(θ)+24=1\sin^2(\theta) + \frac{2}{4} = 1

sin⁡2(θ)+12=1\sin^2(\theta) + \frac{1}{2} = 1

Now, we solve for sin⁡2(θ)\sin^2(\theta):

sin⁡2(θ)=1−12\sin^2(\theta) = 1 - \frac{1}{2}

sin⁡2(θ)=12\sin^2(\theta) = \frac{1}{2}

Taking the square root of both sides, we get:

sin⁡(θ)=±12\sin(\theta) = \pm \sqrt{\frac{1}{2}}

sin⁡(θ)=±22\sin(\theta) = \pm \frac{\sqrt{2}}{2}

Here's where the quadrant information becomes vital. Since θ\theta is in the fourth quadrant, sine is negative. Therefore, we choose the negative root:

sin⁡(θ)=−22\sin(\theta) = -\frac{\sqrt{2}}{2}

This result aligns perfectly with our understanding of trigonometric functions in the fourth quadrant. The negative sign is not just a mathematical artifact; it's a direct consequence of the angle's location. It's a testament to the quadrant's influence on the sign of the sine function. This careful consideration of the quadrant is what separates a correct solution from an incorrect one. Always remember to interpret the signs of trigonometric functions based on their respective quadrants.

Determining Tangent (tan(θ))

Now that we have both sin⁡(θ)\sin(\theta) and cos⁡(θ)\cos(\theta), we can easily find tan⁡(θ)\tan(\theta) using the following identity:

tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}

Substituting the values we found:

tan⁡(θ)=−2222\tan(\theta) = \frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}

tan⁡(θ)=−1\tan(\theta) = -1

The tangent value is negative, which is consistent with our knowledge that tangent is negative in the fourth quadrant. This serves as a good check on our work, reinforcing the importance of quadrant analysis. The tangent function, representing the ratio of sine to cosine, elegantly combines the information we've gathered. Its negative value is a direct consequence of the opposing signs of sine and cosine in the fourth quadrant. This simple calculation encapsulates the core relationship between these three fundamental trigonometric functions. Understanding this relationship and the impact of the quadrant on their signs is crucial for mastering trigonometry.

Summarizing the Results

In conclusion, given cos⁡(θ)=22\cos(\theta) = \frac{\sqrt{2}}{2} and 3π2<θ<2π\frac{3\pi}{2} < \theta < 2\pi, we have successfully evaluated:

  • sin⁡(θ)=−22\sin(\theta) = -\frac{\sqrt{2}}{2}
  • tan⁡(θ)=−1\tan(\theta) = -1

This problem demonstrates the importance of using both trigonometric identities and quadrant information to determine the values of trigonometric functions. By understanding the relationships between sine, cosine, and tangent, and by paying close attention to the quadrant in which the angle lies, we can accurately solve a wide range of trigonometric problems. The process we've undertaken highlights the interconnectedness of trigonometric concepts. Each step, from identifying the reference angle to considering the quadrant, builds upon the previous one. This holistic approach is key to success in trigonometry.

This exercise not only provides specific answers but also reinforces a general problem-solving strategy in trigonometry. It emphasizes the need for a methodical approach, combining algebraic manipulation with geometric understanding. The quadrant serves as a filter, allowing us to select the correct sign for the trigonometric functions. This nuanced understanding is what elevates problem-solving from mere calculation to true mathematical insight.

Final Answers:

tan⁡(θ)=−1\tan(\theta) = -1

sin⁡(θ)=−22\sin(\theta) = -\frac{\sqrt{2}}{2}