Find Cos ⁡ Θ \cos \theta Cos Θ If Sin ⁡ Θ = − 3 13 13 \sin \theta = -\frac{3 \sqrt{13}}{13} Sin Θ = − 13 3 13 ​ ​ And { \theta$}$ Is In The Fourth Quadrant.A. 2 13 13 \frac{2 \sqrt{13}}{13} 13 2 13 ​ ​ B. − 3 2 -\frac{3}{2} − 2 3 ​ C. − 2 13 13 -\frac{2 \sqrt{13}}{13} − 13 2 13 ​ ​ D.

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Introduction

In trigonometry, the sine and cosine functions are fundamental components of the unit circle. Given the value of sine, we can find the value of cosine using the Pythagorean identity. In this article, we will explore how to find the cosine of an angle in the fourth quadrant when the sine value is known.

The Pythagorean Identity

The Pythagorean identity states that for any angle θ\theta, the following equation holds:

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

This identity can be used to find the cosine of an angle when the sine value is known.

Given Information

We are given that sin⁡θ=−31313\sin \theta = -\frac{3 \sqrt{13}}{13} and that θ\theta is in the fourth quadrant. We need to find the value of cos⁡θ\cos \theta.

Finding Cosine

Using the Pythagorean identity, we can find the cosine of the angle as follows:

cos⁡2θ=1−sin⁡2θ\cos^2 \theta = 1 - \sin^2 \theta

Substituting the given value of sine, we get:

cos⁡2θ=1−(−31313)2\cos^2 \theta = 1 - \left(-\frac{3 \sqrt{13}}{13}\right)^2

Simplifying the expression, we get:

cos⁡2θ=1−9⋅13132\cos^2 \theta = 1 - \frac{9 \cdot 13}{13^2}

cos⁡2θ=1−913\cos^2 \theta = 1 - \frac{9}{13}

cos⁡2θ=413\cos^2 \theta = \frac{4}{13}

Taking the square root of both sides, we get:

cos⁡θ=±413\cos \theta = \pm \sqrt{\frac{4}{13}}

Since the angle θ\theta is in the fourth quadrant, the cosine value will be negative. Therefore, we take the negative square root:

cos⁡θ=−413\cos \theta = -\sqrt{\frac{4}{13}}

Simplifying the expression, we get:

cos⁡θ=−213\cos \theta = -\frac{2}{\sqrt{13}}

Rationalizing the denominator, we get:

cos⁡θ=−21313\cos \theta = -\frac{2 \sqrt{13}}{13}

Conclusion

In this article, we found the value of cosine in the fourth quadrant when the sine value is known. We used the Pythagorean identity to find the cosine value and simplified the expression to get the final answer.

Answer

Introduction

In our previous article, we explored how to find the cosine of an angle in the fourth quadrant when the sine value is known. In this article, we will answer some frequently asked questions related to finding cosine in the fourth quadrant.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental concept in trigonometry that states that for any angle θ\theta, the following equation holds:

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

This identity can be used to find the cosine of an angle when the sine value is known.

Q: How do I find the cosine of an angle in the fourth quadrant?

A: To find the cosine of an angle in the fourth quadrant, you can use the Pythagorean identity. First, find the square of the sine value, then subtract it from 1. Take the square root of the result and make sure to take the negative square root since the angle is in the fourth quadrant.

Q: What if the sine value is negative?

A: If the sine value is negative, it means that the angle is in the third or fourth quadrant. To find the cosine value, you can use the Pythagorean identity as before. However, make sure to take the negative square root since the angle is in the third or fourth quadrant.

Q: Can I use the Pythagorean identity to find the sine value?

A: Yes, you can use the Pythagorean identity to find the sine value. If you know the cosine value, you can rearrange the identity to solve for the sine value.

Q: What if I have a decimal value for the sine value?

A: If you have a decimal value for the sine value, you can use a calculator to find the cosine value. Alternatively, you can use the Pythagorean identity to find the cosine value.

Q: Can I use the Pythagorean identity to find the cosine value of a sum or difference of angles?

A: No, the Pythagorean identity only works for a single angle. To find the cosine value of a sum or difference of angles, you need to use the angle addition and subtraction formulas.

Q: What are some common mistakes to avoid when finding the cosine value?

A: Some common mistakes to avoid when finding the cosine value include:

  • Not taking the negative square root when the angle is in the third or fourth quadrant
  • Not using the Pythagorean identity correctly
  • Not simplifying the expression correctly
  • Not rationalizing the denominator when necessary

Conclusion

In this article, we answered some frequently asked questions related to finding cosine in the fourth quadrant. We hope that this article has been helpful in clarifying any doubts you may have had.

Additional Resources

If you want to learn more about trigonometry and the Pythagorean identity, we recommend checking out the following resources:

  • Khan Academy: Trigonometry
  • MIT OpenCourseWare: Trigonometry
  • Wolfram MathWorld: Pythagorean Identity

We hope that this article has been helpful in your journey to learn trigonometry. Good luck!