Find The Exact Value Of $g(2 \theta$\], If $g(\theta) = \cos \theta$, And $\theta = \pi$.

Introduction

In trigonometry, we often encounter functions that involve the cosine of an angle. One such function is $g(\theta) = \cos \theta$, where $\theta$ is the angle. In this article, we will explore how to find the exact value of $g(2\theta)$, given that $\theta = \pi$. We will use the properties of trigonometric functions and identities to derive the solution.

Understanding the Function $g(\theta)$

The function $g(\theta) = \cos \theta$ is a basic trigonometric function that represents the cosine of an angle $\theta$. The cosine function is a periodic function that oscillates between -1 and 1. It is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle.

The Angle $\theta = \pi$

In this problem, we are given that $\theta = \pi$. The value of $\pi$ is approximately 3.14159, but we will work with the exact value in this solution. The angle $\pi$ is a special angle in trigonometry, and its cosine value is well-known.

Finding the Value of $g(2\theta)$

To find the value of $g(2\theta)$, we need to substitute $2\theta$ into the function $g(\theta) = \cos \theta$. This gives us $g(2\theta) = \cos 2\theta$. We can use the double-angle identity for cosine to simplify this expression.

Double-Angle Identity for Cosine

The double-angle identity for cosine states that $\cos 2\theta = 2\cos^2 \theta - 1$. We can use this identity to rewrite the expression $g(2\theta) = \cos 2\theta$.

Substituting $\theta = \pi$ into the Double-Angle Identity

Now that we have the double-angle identity for cosine, we can substitute $\theta = \pi$ into the expression. This gives us $\cos 2\pi = 2\cos^2 \pi - 1$.

Evaluating the Expression

To evaluate the expression $\cos 2\pi = 2\cos^2 \pi - 1$, we need to find the value of $\cos \pi$. The value of $\cos \pi$ is well-known, and it is equal to -1.

Substituting $\cos \pi = -1$ into the Expression

Now that we have the value of $\cos \pi$, we can substitute it into the expression $\cos 2\pi = 2\cos^2 \pi - 1$. This gives us $\cos 2\pi = 2(-1)^2 - 1$.

Evaluating the Final Expression

To evaluate the final expression $\cos 2\pi = 2(-1)^2 - 1$, we need to simplify the expression. This gives us $\cos 2\pi = 2(1) - 1 = 1$.

Conclusion

In this article, we have found the exact value of $g(2\theta)$, given that $\theta = \pi$. We used the properties of trigonometric functions and identities to derive the solution. The final answer is $\boxed{1}$.

Final Answer

The final answer is $\boxed{1}$.

Additional Resources

If you are interested in learning more about trigonometric functions and identities, I recommend checking out the following resources:

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

These resources provide a comprehensive introduction to trigonometry and its applications.

Frequently Asked Questions

Q: What is the value of $\cos \pi$? A: The value of $\cos \pi$ is -1.

Q: What is the double-angle identity for cosine? A: The double-angle identity for cosine is $\cos 2\theta = 2\cos^2 \theta - 1$.

Q: How do I find the value of $g(2\theta)$, given that $\theta = \pi$? A: To find the value of $g(2\theta)$, you need to substitute $2\theta$ into the function $g(\theta) = \cos \theta$. Then, you can use the double-angle identity for cosine to simplify the expression.

Glossary

• Trigonometric function: A function that involves the sine, cosine, or tangent of an angle.
• Double-angle identity: An identity that relates the cosine of an angle to the cosine of twice the angle.
• Cosine function: A trigonometric function that represents the cosine of an angle.
• Angle: A measure of the amount of rotation between two lines or planes.
  

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and computer science. In this article, we will address some of the most frequently asked questions about trigonometry and provide detailed answers to help you better understand the subject.

Q&A

Q: What is trigonometry?

A: Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It involves the study of triangles, particularly right triangles, and the relationships between their sides and angles.

Q: What are the basic trigonometric functions?

A: The basic trigonometric functions are:

• Sine (sin): the ratio of the length of the side opposite the angle to the length of the hypotenuse
• Cosine (cos): the ratio of the length of the side adjacent to the angle to the length of the hypotenuse
• Tangent (tan): the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle

Q: What is the difference between sine, cosine, and tangent?

A: The main difference between sine, cosine, and tangent is the ratio of the sides of the triangle that they represent. Sine represents the ratio of the opposite side to the hypotenuse, cosine represents the ratio of the adjacent side to the hypotenuse, and tangent represents the ratio of the opposite side to the adjacent side.

Q: What is the double-angle identity for sine?

A: The double-angle identity for sine is:

sin(2θ) = 2sin(θ)cos(θ)

Q: What is the double-angle identity for cosine?

A: The double-angle identity for cosine is:

cos(2θ) = 2cos^2(θ) - 1

Q: How do I find the value of sin(θ) if I know the value of cos(θ)?

A: To find the value of sin(θ) if you know the value of cos(θ), you can use the Pythagorean identity:

sin^2(θ) + cos^2(θ) = 1

Rearranging this equation, you get:

sin(θ) = ±√(1 - cos^2(θ))

Q: What is the value of sin(π/2)?

A: The value of sin(π/2) is 1.

Q: What is the value of cos(π/2)?

A: The value of cos(π/2) is 0.

Q: How do I find the value of tan(θ) if I know the value of sin(θ) and cos(θ)?

A: To find the value of tan(θ) if you know the value of sin(θ) and cos(θ), you can use the following formula:

tan(θ) = sin(θ) / cos(θ)

Q: What is the value of tan(π/4)?

A: The value of tan(π/4) is 1.

Q: What is the value of cot(θ)?

A: The value of cot(θ) is the reciprocal of tan(θ), or:

cot(θ) = 1 / tan(θ)

Q: What is the value of sec(θ)?

A: The value of sec(θ) is the reciprocal of cos(θ), or:

sec(θ) = 1 / cos(θ)

Q: What is the value of csc(θ)?

A: The value of csc(θ) is the reciprocal of sin(θ), or:

csc(θ) = 1 / sin(θ)

Conclusion

In this article, we have addressed some of the most frequently asked questions about trigonometry and provided detailed answers to help you better understand the subject. We hope that this article has been helpful in clarifying any doubts you may have had about trigonometry.

Additional Resources

If you are interested in learning more about trigonometry, we recommend checking out the following resources:

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

These resources provide a comprehensive introduction to trigonometry and its applications.

Glossary

• Trigonometric function: A function that involves the sine, cosine, or tangent of an angle.
• Double-angle identity: An identity that relates the cosine of an angle to the cosine of twice the angle.
• Cosine function: A trigonometric function that represents the cosine of an angle.
• Angle: A measure of the amount of rotation between two lines or planes.
• Hypotenuse: The side of a right triangle opposite the right angle.
• Opposite side: The side of a right triangle opposite the angle being considered.
• Adjacent side: The side of a right triangle adjacent to the angle being considered.
• Reciprocal: A number that is the inverse of another number.