Find Cos(θ) & Tan(θ) If Sin(θ) = √5/5 (Quadrant II)

Hey guys! Let's dive into a fun trig problem where we need to figure out the exact values of $\cos(\theta)$ and $\tan(\theta)$, knowing that $\sin(\theta) = \frac{\sqrt{5}}{5}$ and that our angle $\theta$ is chilling in Quadrant II. Buckle up, it’s gonna be a trigonometric ride!

Understanding the Basics

Before we roll up our sleeves, let's refresh some key concepts. We're going to heavily lean on the Pythagorean identity and the definition of the tangent function. Also, knowing the properties of quadrants is super important.

The Pythagorean Identity

The Pythagorean identity is a cornerstone in trigonometry. It states:

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

This magical formula relates the sine and cosine of any angle and is derived directly from the Pythagorean theorem applied to the unit circle. It's our starting point to find $\cos(\theta)$.

Tangent Definition

The tangent function, $\tan(\theta)$, is defined as the ratio of the sine to the cosine:

$$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$$

Once we find $\cos(\theta)$, we can easily compute $\tan(\theta)$ using this definition.

Quadrant Awareness

Knowing which quadrant our angle $\theta$ lies in is crucial because it tells us about the signs of sine, cosine, and tangent. Here’s a quick rundown:

• Quadrant I: All trig functions are positive.
• Quadrant II: Sine is positive, cosine and tangent are negative.
• Quadrant III: Tangent is positive, sine and cosine are negative.
• Quadrant IV: Cosine is positive, sine and tangent are negative.

Since $\theta$ is in Quadrant II, we know that $\sin(\theta)$ is positive (which we already know), $\cos(\theta)$ will be negative, and consequently, $\tan(\theta)$ will also be negative. Keep this in mind as we proceed!

Finding Cos(θ)

Okay, let's use the Pythagorean identity to find $\cos(\theta)$. We know that $\sin(\theta) = \frac{\sqrt{5}}{5}$, so let’s plug that in:

$$\left(\frac{\sqrt{5}}{5}\right)^2 + \cos^2(\theta) = 1$$

Now, simplify the equation:

$$\frac{5}{25} + \cos^2(\theta) = 1$$

$$\frac{1}{5} + \cos^2(\theta) = 1$$

Subtract $\frac{1}{5}$ from both sides:

$$\cos^2(\theta) = 1 - \frac{1}{5}$$

$$\cos^2(\theta) = \frac{4}{5}$$

Now, take the square root of both sides:

$$\cos(\theta) = \pm\sqrt{\frac{4}{5}}$$

$$\cos(\theta) = \pm\frac{2}{\sqrt{5}}$$

Rationalize the denominator:

$$\cos(\theta) = \pm\frac{2\sqrt{5}}{5}$$

Remember that $\theta$ is in Quadrant II, where cosine is negative. Therefore:

$$\cos(\theta) = -\frac{2\sqrt{5}}{5}$$

Calculating Tan(θ)

Now that we have both $\sin(\theta)$ and $\cos(\theta)$, we can find $\tan(\theta)$ using its definition:

$$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$$

Plug in the values:

$$\tan(\theta) = \frac{\frac{\sqrt{5}}{5}}{-\frac{2\sqrt{5}}{5}}$$

Simplify the fraction:

$$\tan(\theta) = \frac{\sqrt{5}}{5} \cdot \frac{5}{-2\sqrt{5}}$$

$$\tan(\theta) = \frac{1}{-2}$$

$$\tan(\theta) = -\frac{1}{2}$$

So, we found that $\tan(\theta) = -\frac{1}{2}$. This aligns with our understanding that tangent should be negative in Quadrant II.

Final Answer

Alright, let’s wrap it up! Given that $\sin(\theta) = \frac{\sqrt{5}}{5}$ and $\theta$ is in Quadrant II, we’ve successfully found:

$$\cos(\theta) = -\frac{2\sqrt{5}}{5}$$

$$\tan(\theta) = -\frac{1}{2}$$

I hope this breakdown helps you nail similar problems in the future. Keep up the great work, and happy trig-solving!

Key Takeaways

• Use the Pythagorean identity to relate sine and cosine.
• Remember the signs of trig functions in different quadrants.
• Apply the definition of tangent as the ratio of sine to cosine.

Practice Problems

To solidify your understanding, try these practice problems:

1. Find $\cos(\theta)$ and $\tan(\theta)$ if $\sin(\theta) = \frac{3}{5}$ and $\theta$ is in Quadrant I.
2. Find $\sin(\theta)$ and $\tan(\theta)$ if $\cos(\theta) = -\frac{12}{13}$ and $\theta$ is in Quadrant III.
3. Find $\sin(\theta)$ and $\cos(\theta)$ if $\tan(\theta) = \frac{5}{12}$ and $\theta$ is in Quadrant III.

Hints:

• For problem 1, everything will be positive.
• For problem 2, both sine and tangent will be negative.
• For problem 3, you'll need to use another identity to relate tangent to sine and cosine, such as $1 + \tan^2(\theta) = \sec^2(\theta)$.

Additional Tips

1. Draw Diagrams: Sketching the angle in the correct quadrant can give you a visual check on the signs of your answers.
2. Memorize Identities: The more you use the basic trig identities, the easier they will become to recall.
3. Check Your Work: Always ensure your final answers make sense in the context of the given quadrant. This can help you catch sign errors.

Conclusion

In conclusion, mastering trigonometric identities and understanding quadrant properties are fundamental skills in trigonometry. By systematically applying these concepts, you can solve complex problems step by step. Remember, practice makes perfect, so keep solving problems, and you'll become more confident and proficient in no time. Keep exploring, keep learning, and most importantly, have fun with math! You got this!