Evaluate Tan(θ) If Cos(θ) = √2/2 And 3π/2 < Θ < 2π

Hey everyone! Let's dive into a trigonometric problem where we need to find the value of tan(θ) given some information about cos(θ) and the range of θ. This is a classic problem that combines our understanding of trigonometric functions and their properties in different quadrants. So, let's get started and break it down step by step.

Understanding the Problem

In this problem, our main goal is to determine the value of tan(θ). We are given that cos(θ) = √2/2, and θ lies in the interval (3π/2, 2π). This interval is crucial because it tells us which quadrant θ is in, which will affect the sign of our trigonometric functions. Remember, the unit circle is our friend here! It helps us visualize the signs of sine, cosine, and tangent in each quadrant. In the fourth quadrant (3π/2 < θ < 2π), cosine is positive, sine is negative, and consequently, tangent is negative. This is super important because it narrows down our possible answers and helps us avoid making mistakes with signs.

Before we jump into the calculations, let's quickly recap the basics. The cosine function, cos(θ), represents the x-coordinate of a point on the unit circle, while the sine function, sin(θ), represents the y-coordinate. The tangent function, tan(θ), is defined as sin(θ)/cos(θ). These fundamental definitions are key to solving this problem. So, with this foundation in place, we are well-equipped to tackle the challenge and find the value of tan(θ). Let's move on to the next section where we'll use these concepts to find sin(θ) and then tan(θ).

Finding sin(θ)

Alright, so we know cos(θ) = √2/2, and we need to find sin(θ). How do we do that? Well, the Pythagorean identity is our best friend here! This identity, which states that sin²(θ) + cos²(θ) = 1, is a fundamental relationship between sine and cosine that holds true for any angle θ. It's like a superpower for solving trig problems! We can use this identity to find sin(θ) because we already know cos(θ). Let's plug in the value of cos(θ) into the identity:

sin²(θ) + (√2/2)² = 1

sin²(θ) + 2/4 = 1

sin²(θ) = 1 - 2/4

sin²(θ) = 2/4

sin²(θ) = 1/2

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

sin(θ) = ±√(1/2)

sin(θ) = ±√2/2

Okay, we have two possible values for sin(θ): √2/2 and -√2/2. But remember, we know that θ is in the fourth quadrant (3π/2 < θ < 2π). In the fourth quadrant, sine is negative. So, we can confidently say that sin(θ) = -√2/2. This is a crucial step, guys! We've successfully found the value of sin(θ) by using the Pythagorean identity and considering the quadrant in which θ lies. Now that we have both sin(θ) and cos(θ), we're just one step away from finding tan(θ). Let's move on to the final calculation!

Evaluating tan(θ)

Now that we know sin(θ) = -√2/2 and cos(θ) = √2/2, we can finally find tan(θ). Remember, tangent is defined as the ratio of sine to cosine: tan(θ) = sin(θ)/cos(θ). This is a super straightforward calculation now that we have both values. Let's plug in the values of sin(θ) and cos(θ) into the formula:

tan(θ) = (-√2/2) / (√2/2)

This looks simple enough, right? We're dividing a value by itself (but with a negative sign), so the result will be -1. Let's simplify:

tan(θ) = -1

And there we have it! We've successfully evaluated tan(θ). The value of tan(θ) is -1. This makes perfect sense, considering that θ is in the fourth quadrant where tangent is negative. We've used the given information about cos(θ), the range of θ, and the fundamental trigonometric identities to arrive at this answer. This is a great example of how all the pieces of trigonometry fit together! So, now you know how to tackle these types of problems. Let's recap the key steps we took to solve this problem.

Conclusion

So, let's wrap things up and recap what we've done to evaluate tan(θ). We started with the given information: cos(θ) = √2/2 and 3π/2 < θ < 2π. Our goal was to find the value of tan(θ). The first key step was to recognize that the interval 3π/2 < θ < 2π places θ in the fourth quadrant, where cosine is positive and sine is negative. This understanding helped us determine the correct sign for sin(θ).

Next, we used the Pythagorean identity, sin²(θ) + cos²(θ) = 1, to find sin(θ). By plugging in the value of cos(θ) and solving for sin(θ), we found two possible values: √2/2 and -√2/2. However, since we knew θ was in the fourth quadrant, we chose the negative value, sin(θ) = -√2/2.

Finally, we used the definition of tangent, tan(θ) = sin(θ)/cos(θ), to calculate tan(θ). By plugging in the values we found for sin(θ) and cos(θ), we got tan(θ) = (-√2/2) / (√2/2) = -1. This result aligns perfectly with our understanding that tangent is negative in the fourth quadrant.

This problem is a great example of how trigonometric identities and the unit circle work together to solve for unknown values. By breaking down the problem into smaller steps and carefully considering the signs of trigonometric functions in different quadrants, we were able to find the solution. Remember, practice makes perfect, guys! The more you work through these types of problems, the more comfortable you'll become with trigonometry. Keep up the great work, and you'll be trig masters in no time!

If $\cos (\theta)=\frac{\sqrt{2}}{2}$, and $rac{3 \pi}{2}\ \textless \ \theta\ \textless \ 2 \pi$, evaluate $\tan (\theta)$.