Let $ F(x) = \sin X $, $ G(x) = \cos X $, And $ H(x) = 2x $. Find The Value Of The Following: $ (f+g)\left(\frac{\pi}{3}\right) $

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Introduction

In this article, we will explore the concept of function addition and evaluate the value of the expression $ (f+g)\left(\frac{\pi}{3}\right) $, where $ f(x) = \sin x $, $ g(x) = \cos x $, and $ h(x) = 2x $. We will start by understanding the concept of function addition and then proceed to evaluate the given expression.

Understanding Function Addition

Function addition is a fundamental concept in mathematics that involves combining two or more functions to form a new function. Given two functions $ f(x) $ and $ g(x) $, the sum of these functions is defined as $ (f+g)(x) = f(x) + g(x) $. This means that for any input value $ x $, the output of the sum function is the sum of the outputs of the individual functions.

Evaluating the Expression

To evaluate the expression $ (f+g)\left(\frac{\pi}{3}\right) $, we need to find the values of $ f\left(\frac{\pi}{3}\right) $ and $ g\left(\frac{\pi}{3}\right) $ and then add them together.

Evaluating $ f\left(\frac{\pi}{3}\right) $

The function $ f(x) = \sin x $ is the sine function, which is a periodic function that oscillates between -1 and 1. To evaluate $ f\left(\frac{\pi}{3}\right) $, we need to find the value of the sine function at $ x = \frac{\pi}{3} $.

Using the unit circle or a trigonometric table, we can find that $ \sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} $.

Evaluating $ g\left(\frac{\pi}{3}\right) $

The function $ g(x) = \cos x $ is the cosine function, which is also a periodic function that oscillates between -1 and 1. To evaluate $ g\left(\frac{\pi}{3}\right) $, we need to find the value of the cosine function at $ x = \frac{\pi}{3} $.

Using the unit circle or a trigonometric table, we can find that $ \cos \left(\frac{\pi}{3}\right) = \frac{1}{2} $.

Adding the Values

Now that we have found the values of $ f\left(\frac{\pi}{3}\right) $ and $ g\left(\frac{\pi}{3}\right) $, we can add them together to find the value of $ (f+g)\left(\frac{\pi}{3}\right) $.

$ (f+g)\left(\frac{\pi}{3}\right) = f\left(\frac{\pi}{3}\right) + g\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} + \frac{1}{2} $

Simplifying the Expression

To simplify the expression, we can combine the fractions by finding a common denominator.

$ \frac{\sqrt{3}}{2} + \frac{1}{2} = \frac{\sqrt{3} + 1}{2} $

Conclusion

In this article, we have evaluated the expression $ (f+g)\left(\frac{\pi}{3}\right) $, where $ f(x) = \sin x $, $ g(x) = \cos x $, and $ h(x) = 2x $. We have found that the value of the expression is $ \frac{\sqrt{3} + 1}{2} $. This demonstrates the concept of function addition and how it can be used to evaluate complex expressions.

Final Answer

The final answer is: 3+12\boxed{\frac{\sqrt{3} + 1}{2}}

Introduction

In our previous article, we evaluated the expression $ (f+g)\left(\frac{\pi}{3}\right) $, where $ f(x) = \sin x $, $ g(x) = \cos x $, and $ h(x) = 2x $. We found that the value of the expression is $ \frac{\sqrt{3} + 1}{2} $. In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q: What is the concept of function addition?

A: Function addition is a fundamental concept in mathematics that involves combining two or more functions to form a new function. Given two functions $ f(x) $ and $ g(x) $, the sum of these functions is defined as $ (f+g)(x) = f(x) + g(x) $.

Q: How do you evaluate the expression $ (f+g)\left(\frac{\pi}{3}\right) $?

A: To evaluate the expression $ (f+g)\left(\frac{\pi}{3}\right) $, you need to find the values of $ f\left(\frac{\pi}{3}\right) $ and $ g\left(\frac{\pi}{3}\right) $ and then add them together.

Q: What is the value of $ f\left(\frac{\pi}{3}\right) $?

A: The value of $ f\left(\frac{\pi}{3}\right) $ is $ \frac{\sqrt{3}}{2} $, which is the value of the sine function at $ x = \frac{\pi}{3} $.

Q: What is the value of $ g\left(\frac{\pi}{3}\right) $?

A: The value of $ g\left(\frac{\pi}{3}\right) $ is $ \frac{1}{2} $, which is the value of the cosine function at $ x = \frac{\pi}{3} $.

Q: How do you add the values of $ f\left(\frac{\pi}{3}\right) $ and $ g\left(\frac{\pi}{3}\right) $?

A: To add the values of $ f\left(\frac{\pi}{3}\right) $ and $ g\left(\frac{\pi}{3}\right) $, you can combine the fractions by finding a common denominator.

Q: What is the final answer to the expression $ (f+g)\left(\frac{\pi}{3}\right) $?

A: The final answer to the expression $ (f+g)\left(\frac{\pi}{3}\right) $ is $ \frac{\sqrt{3} + 1}{2} $.

Conclusion

In this article, we have answered some frequently asked questions related to the expression $ (f+g)\left(\frac{\pi}{3}\right) $. We hope that this article has provided a clear understanding of the concept of function addition and how it can be used to evaluate complex expressions.

Final Answer

The final answer is: 3+12\boxed{\frac{\sqrt{3} + 1}{2}}

Related Questions

  • What is the concept of function composition?
  • How do you evaluate the expression $ (f\circ g)\left(x\right) $?
  • What is the value of $ f\left(g\left(x\right)\right) $?
  • How do you add the values of $ f\left(g\left(x\right)\right) $ and $ g\left(f\left(x\right)\right) $?
  • What is the final answer to the expression $ (f\circ g)\left(x\right) $?

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