Finding Cosine of the Angle

Oct 13, 2025 by miodeya.com 138 views

Introduction

In trigonometry, the double angle formula for cosine is a fundamental concept used to find the cosine of a double angle. Given the sine of an angle, we can use the Pythagorean identity to find the cosine of the angle. Then, we can use the double angle formula to find the cosine of the double angle. In this article, we will discuss how to find the cosine of a double angle given the sine of the original angle.

The Pythagorean Identity

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

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

We can use this identity to find the cosine of the angle given the sine of the angle.

Finding Cosine of the Angle

Given that $\sin \theta = \frac{20}{29}$ , we can use the Pythagorean identity to find the cosine of the angle. We have:

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

Substituting the value of $\sin \theta$ , we get:

\left(\frac{20}{29}\right)^2 + \cos^2 \theta = 1

Simplifying the equation, we get:

\frac{400}{841} + \cos^2 \theta = 1

Subtracting $\frac{400}{841}$ from both sides, we get:

\cos^2 \theta = \frac{441}{841}

Since $\cos \theta > 0$ , we can take the square root of both sides to get:

\cos \theta = \frac{21}{29}

The Double Angle Formula

The double angle formula for cosine states that for any angle $\theta$ , the following equation holds:

\cos (2 \theta) = 2 \cos^2 \theta - 1

We can use this formula to find the cosine of the double angle.

Finding Cosine of the Double Angle

Substituting the value of $\cos \theta$ into the double angle formula, we get:

\cos (2 \theta) = 2 \left(\frac{21}{29}\right)^2 - 1

Simplifying the equation, we get:

\cos (2 \theta) = 2 \left(\frac{441}{841}\right) - 1

Simplifying further, we get:

\cos (2 \theta) = \frac{882}{841} - 1

Subtracting 1 from both sides, we get:

\cos (2 \theta) = \frac{882}{841} - \frac{841}{841}

Simplifying further, we get:

\cos (2 \theta) = \frac{41}{841}

Conclusion

In this article, we discussed how to find the cosine of a double angle given the sine of the original angle. We used the Pythagorean identity to find the cosine of the angle, and then used the double angle formula to find the cosine of the double angle. We found that the cosine of the double angle is $\frac{41}{841}$ .

Answer

The correct answer is D) $\frac{41}{841}$ .

Discussion

This problem is a classic example of how to use the Pythagorean identity and the double angle formula to find the cosine of a double angle. The problem requires the student to use algebraic manipulations to simplify the equation and find the final answer. The student must also be able to identify the correct answer from the given options.

Tips and Tricks

Make sure to use the Pythagorean identity to find the cosine of the angle before using the double angle formula.
Use algebraic manipulations to simplify the equation and find the final answer.
Identify the correct answer from the given options.

References

"Trigonometry" by Michael Corral
"Precalculus" by James Stewart
"Calculus" by Michael Spivak
Q&A: Finding Cosine of Double Angle in Trigonometry =====================================================

Q: What is the Pythagorean identity?

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

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

This identity can be used to find the cosine of an angle given the sine of the angle.

Q: How do I use the Pythagorean identity to find the cosine of an angle?

A: To use the Pythagorean identity to find the cosine of an angle, you need to substitute the value of the sine of the angle into the equation and solve for the cosine of the angle. For example, if $\sin \theta = \frac{20}{29}$ , you can use the Pythagorean identity to find the cosine of the angle as follows:

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

Substituting the value of $\sin \theta$ , we get:

\left(\frac{20}{29}\right)^2 + \cos^2 \theta = 1

Simplifying the equation, we get:

\frac{400}{841} + \cos^2 \theta = 1

Subtracting $\frac{400}{841}$ from both sides, we get:

\cos^2 \theta = \frac{441}{841}

Since $\cos \theta > 0$ , we can take the square root of both sides to get:

\cos \theta = \frac{21}{29}

Q: What is the double angle formula for cosine?

A: The double angle formula for cosine states that for any angle $\theta$ , the following equation holds:

\cos (2 \theta) = 2 \cos^2 \theta - 1

This formula can be used to find the cosine of a double angle given the cosine of the original angle.

Q: How do I use the double angle formula to find the cosine of a double angle?

A: To use the double angle formula to find the cosine of a double angle, you need to substitute the value of the cosine of the original angle into the equation and solve for the cosine of the double angle. For example, if $\cos \theta = \frac{21}{29}$ , you can use the double angle formula to find the cosine of the double angle as follows:

\cos (2 \theta) = 2 \cos^2 \theta - 1

Substituting the value of $\cos \theta$ , we get:

\cos (2 \theta) = 2 \left(\frac{21}{29}\right)^2 - 1

Simplifying the equation, we get:

\cos (2 \theta) = 2 \left(\frac{441}{841}\right) - 1

Simplifying further, we get:

\cos (2 \theta) = \frac{882}{841} - 1

Subtracting 1 from both sides, we get:

\cos (2 \theta) = \frac{882}{841} - \frac{841}{841}

Simplifying further, we get:

\cos (2 \theta) = \frac{41}{841}

Q: What are some common mistakes to avoid when finding the cosine of a double angle?

A: Some common mistakes to avoid when finding the cosine of a double angle include:

Not using the Pythagorean identity to find the cosine of the original angle before using the double angle formula.
Not simplifying the equation correctly when using the double angle formula.
Not checking the sign of the cosine of the double angle.

Q: How can I practice finding the cosine of a double angle?

A: You can practice finding the cosine of a double angle by working through examples and exercises in a trigonometry textbook or online resource. You can also try using online calculators or software to check your answers and get feedback on your work.

Q: What are some real-world applications of the cosine of a double angle?

A: The cosine of a double angle has many real-world applications, including:

Physics: The cosine of a double angle is used to describe the motion of objects in terms of their position, velocity, and acceleration.
Engineering: The cosine of a double angle is used to design and analyze mechanical systems, such as gears and linkages.
Computer Science: The cosine of a double angle is used in algorithms for image and signal processing.

Q: Can I use the cosine of a double angle to find the sine of a double angle?

A: Yes, you can use the cosine of a double angle to find the sine of a double angle. Since $\sin^2 \theta + \cos^2 \theta = 1$ , you can use the cosine of a double angle to find the sine of a double angle as follows:

\sin (2 \theta) = \sqrt{1 - \cos^2 (2 \theta)}

This formula can be used to find the sine of a double angle given the cosine of the double angle.