Find The Value Of The Given Trigonometric Function: Cos ⁡ 420 ∘ \cos 420^{\circ} Cos 42 0 ∘ A. 3 \sqrt{3} 3 ​ B. 2 2 \frac{\sqrt{2}}{2} 2 2 ​ ​ C. 1 2 \frac{1}{2} 2 1 ​ D. 3 2 \frac{\sqrt{3}}{2} 2 3 ​ ​ Please Select The Best Answer From The Choices

Introduction

Trigonometric functions are an essential part of mathematics, and understanding their values is crucial for solving various mathematical problems. In this article, we will focus on finding the value of the given trigonometric function: $\cos 420^{\circ}$. We will explore the different methods to evaluate this function and provide a step-by-step guide to arrive at the correct answer.

Understanding the Trigonometric Function

The trigonometric function we are dealing with is the cosine function, denoted by $\cos$. The cosine function is a periodic function, meaning it repeats its values at regular intervals. The period of the cosine function is $360^{\circ}$, which means that the function repeats its values every $360^{\circ}$.

Evaluating the Trigonometric Function

To evaluate the trigonometric function $\cos 420^{\circ}$, we need to consider the periodic nature of the function. Since the period of the cosine function is $360^{\circ}$, we can subtract multiples of $360^{\circ}$ from the given angle to bring it within the range of $0^{\circ}$ to $360^{\circ}$.

Method 1: Subtracting Multiples of 360°

We can subtract $360^{\circ}$ from $420^{\circ}$ to bring it within the range of $0^{\circ}$ to $360^{\circ}$.

$\cos 420^{\circ} = \cos (420^{\circ} - 360^{\circ})$ $\cos 420^{\circ} = \cos 60^{\circ}$

Method 2: Using the Reference Angle

Another method to evaluate the trigonometric function is to use the reference angle. The reference angle is the acute angle between the terminal side of the angle and the x-axis.

In this case, the reference angle for $420^{\circ}$ is $60^{\circ}$.

$\cos 420^{\circ} = \cos (360^{\circ} + 60^{\circ})$ $\cos 420^{\circ} = \cos 60^{\circ}$

Evaluating the Cosine Function

Now that we have brought the angle within the range of $0^{\circ}$ to $360^{\circ}$, we can evaluate the cosine function.

$\cos 60^{\circ} = \frac{1}{2}$

Conclusion

In conclusion, the value of the given trigonometric function $\cos 420^{\circ}$ is $\frac{1}{2}$. We used two methods to evaluate the function: subtracting multiples of $360^{\circ}$ and using the reference angle. Both methods led us to the same result, which is $\frac{1}{2}$.

Answer

The correct answer is:

• c. $\frac{1}{2}$

Final Thoughts

Introduction

In our previous article, we explored the concept of finding the value of a trigonometric function, specifically the cosine function. We provided a step-by-step guide on how to evaluate the function and arrived at the correct answer. In this article, we will address some common questions and answers related to trigonometric functions.

Q&A

Q: What is the period of the cosine function?

A: The period of the cosine function is $360^{\circ}$, which means that the function repeats its values every $360^{\circ}$.

Q: How do I evaluate a trigonometric function with a negative angle?

A: To evaluate a trigonometric function with a negative angle, you can use the identity $\cos (-x) = \cos x$.

Q: What is the reference angle?

A: The reference angle is the acute angle between the terminal side of the angle and the x-axis.

Q: How do I find the reference angle?

A: To find the reference angle, you can subtract the nearest multiple of $360^{\circ}$ from the given angle.

Q: What is the difference between the sine and cosine functions?

A: The sine function is the ratio of the opposite side to the hypotenuse, while the cosine function is the ratio of the adjacent side to the hypotenuse.

Q: How do I evaluate a trigonometric function with a large angle?

A: To evaluate a trigonometric function with a large angle, you can use the identity $\cos (x + 360^{\circ}) = \cos x$.

Q: What is the value of the cosine function at $0^{\circ}$?

A: The value of the cosine function at $0^{\circ}$ is $1$.

Q: What is the value of the cosine function at $90^{\circ}$?

A: The value of the cosine function at $90^{\circ}$ is $0$.

Q: How do I evaluate a trigonometric function with a complex angle?

A: To evaluate a trigonometric function with a complex angle, you can use the identity $\cos (x + iy) = \cos x \cosh y - i \sin x \sinh y$.

Q: What is the value of the cosine function at $180^{\circ}$?

A: The value of the cosine function at $180^{\circ}$ is $-1$.

Q: What is the value of the cosine function at $270^{\circ}$?

A: The value of the cosine function at $270^{\circ}$ is $0$.

Q: How do I evaluate a trigonometric function with a large angle using a calculator?

A: To evaluate a trigonometric function with a large angle using a calculator, you can use the degree mode and enter the angle in degrees.

Conclusion

In conclusion, trigonometric functions are an essential part of mathematics, and understanding their values is crucial for solving various mathematical problems. By following the steps outlined in this article, you can address common questions and answers related to trigonometric functions and arrive at the correct answer.

Final Thoughts

Trigonometric functions are a fundamental concept in mathematics, and understanding their values is essential for solving various mathematical problems. By following the steps outlined in this article, you can evaluate trigonometric functions with ease and arrive at the correct answer.