Solving Cosine: Finding Sine And Tangent

Hey everyone! Let's dive into a fun trigonometry problem. We're given that $\cos (\theta) = \frac{\sqrt{2}}{2}$, and we know that the angle $\theta$ lies between $\frac{3\pi}{2}$ and $2\pi$. Our mission? To find the values of $\sin (\theta)$ and $\tan (\theta)$. Don't worry, it's not as scary as it sounds. We'll break it down step by step, and you'll be a trig whiz in no time. This problem is a classic example of how to use trigonometric identities and the unit circle to find the values of trigonometric functions when given certain conditions. Let's get started, shall we?

Understanding the Problem

Alright, before we jump into the calculations, let's make sure we understand what the problem is asking. We have a cosine value, which tells us something about the ratio of the adjacent side to the hypotenuse in a right-angled triangle (or, more generally, about the x-coordinate on the unit circle). The fact that $\theta$ is between $\frac{3\pi}{2}$ and $2\pi$ (that's between 270 degrees and 360 degrees, or the fourth quadrant) is super important because it tells us the sign of our sine and tangent values. Remember the acronym CAST? It helps us remember which trig functions are positive in each quadrant:

• Cosine is positive in the fourth quadrant.
• All are positive in the first quadrant.
• Sine is positive in the second quadrant.
• Tangent is positive in the third quadrant.

Since cosine is positive in the fourth quadrant, we know that our cosine value makes sense. However, sine will be negative in this quadrant. This will be very important for when we go to find $\sin (\theta)$. Also, it's super important to remember the Pythagorean identity. The Pythagorean identity is going to allow us to solve for $\sin (\theta)$ given that we know $\cos (\theta)$.

Now, let's get down to the actual math. The most important thing to remember is the Pythagorean Identity: $\sin^2(\theta) + \cos^2(\theta) = 1$. This is our golden ticket. This identity is a fundamental relationship in trigonometry, linking the sine and cosine functions. It's derived directly from the Pythagorean theorem applied to a right triangle inscribed in the unit circle. It’s a must-know for anyone dealing with trig.

Step 1: Finding $\sin(\theta)$

So, using the Pythagorean identity, we can rearrange to solve for $\sin (\theta)$: $\sin^2(\theta) = 1 - \cos^2(\theta)$. Now, plug in the value of $\cos (\theta)$: $\sin^2(\theta) = 1 - (\frac{\sqrt{2}}{2})^2$. Simplify: $\sin^2(\theta) = 1 - \frac{2}{4} = 1 - \frac{1}{2} = \frac{1}{2}$.

Now, to find $\sin(\theta)$, we take the square root of both sides: $\sin(\theta) = \pm \sqrt{\frac{1}{2}} = \pm \frac{\sqrt{2}}{2}$.

But wait! We know that $\theta$ is in the fourth quadrant, where sine is negative. Therefore, $\sin(\theta) = -\frac{\sqrt{2}}{2}$.

Step 2: Finding $\tan(\theta)$

Great! We've found $\sin(\theta)$. Now let's move on to $\tan (\theta)$. Remember the relationship: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$.

Plug in the values we know: $\tan(\theta) = \frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}$.

Simplify: $\tan(\theta) = -1$.

And that's it, guys! We have successfully found both $\sin (\theta)$ and $\tan (\theta)$.

Detailed Explanation

Let's break down each step in a bit more detail, just to make sure everything is crystal clear. We'll explore the use of the Pythagorean identity, and the significance of the quadrant in which the angle lies.

The Pythagorean Identity

As mentioned before, the Pythagorean identity is your best friend in trigonometry. It is based on the Pythagorean theorem ($a^2 + b^2 = c^2$) applied to a right triangle. Consider a right triangle on the unit circle. The legs of the triangle have lengths $\sin(\theta)$ and $\cos(\theta)$, and the hypotenuse has a length of 1. Hence, the Pythagorean identity follows directly from the theorem. This identity is super useful for converting between sine and cosine (or vice versa) when you know one of them and it's also important for simplifying expressions and solving equations. Remember that it's a fundamental concept in trigonometry, so make sure you understand it!

Determining the Correct Sign

The quadrant of $\theta$ ($\frac{3\pi}{2} < \theta < 2\pi$) is super important in this problem. It tells us that the angle is in the fourth quadrant of the unit circle. In this quadrant: cosine is positive, sine is negative, and tangent is negative. This helps us determine the correct sign for our sine and tangent values after using the Pythagorean identity and the definition of tangent. If we didn't consider the quadrant, we might have ended up with incorrect signs for our final answers, which would be a big mistake. The acronym CAST will help you remember this.

Calculating $\tan(\theta)$

Once we have both $\sin(\theta)$ and $\cos(\theta)$, finding $\tan(\theta)$ is straightforward. Just remember the formula $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. This formula is a direct consequence of the definitions of these trigonometric functions using the unit circle or right triangles. Always make sure that you are using the correct signs when plugging in your values, otherwise, your answer will be incorrect. Then just divide and there you have it, you get the tangent value. Easy peasy, right?

Conclusion

So there you have it. We started with $\cos (\theta) = \frac{\sqrt{2}}{2}$ and the information about the quadrant and we were able to find $\sin (\theta) = -\frac{\sqrt{2}}{2}$ and $\tan (\theta) = -1$. Remember that understanding the Pythagorean identity, the unit circle, and the signs of trigonometric functions in each quadrant is key to solving these kinds of problems. Keep practicing and you'll get the hang of it! You will encounter similar questions in your exams and other academic contexts, so mastering these concepts is going to make you succeed. So keep it up, and you'll become a trigonometry master.

Practice Problems

Want to practice more? Try these problems:

1. Given $\sin(\theta) = \frac{1}{2}$ and $0 < \theta < \frac{\pi}{2}$, find $\cos(\theta)$ and $\tan(\theta)$.
2. Given $\cos(\theta) = -\frac{\sqrt{3}}{2}$ and $\pi < \theta < \frac{3\pi}{2}$, find $\sin(\theta)$ and $\tan(\theta)$.

Good luck, and keep learning! You got this! Remember, practice makes perfect. The more problems you solve, the more comfortable you'll become with these concepts.