Finding Sin 2θ Given Cos Θ And Quadrant In Trigonometry

Hey there, math enthusiasts! Today, we're going to unravel a fascinating problem involving trigonometric functions and identities. Specifically, we'll be tackling the challenge of finding the value of $\sin 2\theta$ given that $180^{\circ} < \theta < 270^{\circ}$ and $\cos \theta = -\frac{8}{13}$. This problem is a fantastic example of how understanding the relationships between trigonometric functions and their behavior in different quadrants can lead us to elegant solutions. So, grab your calculators and let's dive in!

Understanding the Problem: Setting the Stage

Before we jump into the calculations, let's make sure we fully grasp the problem at hand. We are given two crucial pieces of information:

1. The angle $\theta$ lies in the third quadrant, as $180^{\circ} < \theta < 270^{\circ}$. This is super important because the signs of trigonometric functions vary depending on the quadrant. Remember the acronym ASTC (All Students Take Calculus) or CAST, which helps us recall which functions are positive in each quadrant:

   • All (all functions are positive) in the first quadrant ($0^{\circ} < \theta < 90^{\circ}$)
   • Sine (and its reciprocal, cosecant) are positive in the second quadrant ($90^{\circ} < \theta < 180^{\circ}$)
   • Tangent (and its reciprocal, cotangent) are positive in the third quadrant ($180^{\circ} < \theta < 270^{\circ}$)
   • Cosine (and its reciprocal, secant) are positive in the fourth quadrant ($270^{\circ} < \theta < 360^{\circ}$)

   In our case, since $\theta$ is in the third quadrant, only tangent and cotangent will be positive. Sine and cosine will be negative, which aligns with the next piece of information.

2. $\cos \theta = -\frac{8}{13}$. The fact that cosine is negative confirms that $\theta$ is indeed in either the second or third quadrant. Combined with the first piece of information, we're certain we're in the third quadrant.

Our goal is to find $\sin 2\theta$. This requires us to use the double-angle formula for sine, which states:

$$\sin 2\theta = 2 \sin \theta \cos \theta$$

So, to find $\sin 2\theta$, we need to determine the value of $\sin \theta$. We already know $\cos \theta$, so our next step is to find $\sin \theta$ using the given information and our knowledge of trigonometric relationships.

Finding $\sin \theta$: Utilizing the Pythagorean Identity

The key to finding $\sin \theta$ when we know $\cos \theta$ is the fundamental Pythagorean identity:

$$\sin^2 \theta + \cos^2 \theta = 1$$

This identity is a cornerstone of trigonometry and allows us to relate sine and cosine. We can rearrange this identity to solve for $\sin \theta$:

$$\sin^2 \theta = 1 - \cos^2 \theta$$

$$\sin \theta = \pm \sqrt{1 - \cos^2 \theta}$$

Notice the $\pm$ sign. This is because the square root function can yield both positive and negative results. However, we know that $\sin \theta$ is negative in the third quadrant, so we will choose the negative root. Now, let's plug in the given value of $\cos \theta = -\frac{8}{13}$:

$$\sin \theta = -\sqrt{1 - \left(-\frac{8}{13}\right)^2}$$

Let's simplify this step-by-step:

$$\sin \theta = -\sqrt{1 - \frac{64}{169}}$$

$$\sin \theta = -\sqrt{\frac{169 - 64}{169}}$$

$$\sin \theta = -\sqrt{\frac{105}{169}}$$

$$\sin \theta = -\frac{\sqrt{105}}{13}$$

Great! We've found $\sin \theta$. Now we have both $\sin \theta$ and $\cos \theta$, and we're ready to tackle the final step.

Calculating $\sin 2\theta$: Applying the Double-Angle Formula

Now that we have $\sin \theta = -\frac{\sqrt{105}}{13}$ and $\cos \theta = -\frac{8}{13}$, we can use the double-angle formula for sine:

$$\sin 2\theta = 2 \sin \theta \cos \theta$$

Let's plug in the values we found:

$$\sin 2\theta = 2 \left(-\frac{\sqrt{105}}{13}\right) \left(-\frac{8}{13}\right)$$

Now, let's multiply:

$$\sin 2\theta = \frac{2 \cdot \sqrt{105} \cdot 8}{13 \cdot 13}$$

$$\sin 2\theta = \frac{16\sqrt{105}}{169}$$

And there we have it! The value of $\sin 2\theta$ is $\frac{16\sqrt{105}}{169}$.

Wrapping Up: Key Takeaways

Let's recap the steps we took to solve this problem:

1. Understood the problem: We identified the given information ($\theta$ in the third quadrant, $\cos \theta = -\frac{8}{13}$) and the goal (find $\sin 2\theta$).
2. Recalled the double-angle formula: We remembered that $\sin 2\theta = 2 \sin \theta \cos \theta$.
3. Found $\sin \theta$: We used the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ and the quadrant information to determine the value of $\sin \theta$.
4. Calculated $\sin 2\theta$: We plugged the values of $\sin \theta$ and $\cos \theta$ into the double-angle formula and simplified.

This problem highlights the importance of understanding trigonometric identities, quadrant rules, and how they all connect. By mastering these concepts, you'll be well-equipped to tackle a wide range of trigonometric challenges. Keep practicing, and you'll become a trig whiz in no time! Remember trigonometry is a fun branch of mathematics that is often used in problem-solving. The double angle formula is one of the keys to solve trigonometric functions.

Practice Problems: Test Your Understanding

To solidify your understanding, try solving these similar problems:

1. If $90^{\circ} < \theta < 180^{\circ}$ and $\sin \theta = \frac{5}{13}$, find $\cos 2\theta$.
2. If $270^{\circ} < \theta < 360^{\circ}$ and $\tan \theta = -\frac{3}{4}$, find $\sin 2\theta$ and $\cos 2\theta$.

Feel free to share your solutions in the comments below! Happy trig-ing, guys! Mastering trigonometric identities and applying them to different scenarios is a valuable skill in mathematics.

Real-World Applications: Where Trigonometry Shines

You might be wondering,