Find All Solutions In \left[0^{\circ}, 360^{\circ}\right ]. 2 Cos ⁡ X − 1 = 0.3326 2 \cos X - 1 = 0.3326 2 Cos X − 1 = 0.3326 X ≈ □ ∘ X \approx \square^{\circ} X ≈ □ ∘ Type Your Answer In Degrees. Do Not Include The Degree Symbol In Your Answer. Round To The Nearest Hundredth As

Introduction

Trigonometric equations are a fundamental aspect of mathematics, and solving them requires a deep understanding of trigonometric functions and their properties. In this article, we will focus on finding solutions to the equation $2 \cos x - 1 = 0.3326$ in the interval $\left[0^{\circ}, 360^{\circ}\right]$. This equation involves the cosine function, which is a periodic function with a period of $360^{\circ}$. We will use various trigonometric identities and properties to solve this equation and find all possible solutions in the given interval.

Understanding the Equation

The given equation is $2 \cos x - 1 = 0.3326$. To solve this equation, we need to isolate the cosine function. We can start by adding 1 to both sides of the equation, which gives us $2 \cos x = 1.3326$. Next, we can divide both sides of the equation by 2, which gives us $\cos x = 0.6663$. This is the equation we need to solve.

Solving the Equation

To solve the equation $\cos x = 0.6663$, we can use the inverse cosine function, denoted by $\cos^{-1}$. The inverse cosine function returns the angle whose cosine is a given value. In this case, we want to find the angle whose cosine is 0.6663. We can use a calculator or a trigonometric table to find this angle.

Finding the Angle

Using a calculator, we can find that the angle whose cosine is 0.6663 is approximately $44.66^{\circ}$. This is one possible solution to the equation $\cos x = 0.6663$. However, we need to find all possible solutions in the interval $\left[0^{\circ}, 360^{\circ}\right]$.

Using Trigonometric Identities

To find all possible solutions, we can use the periodicity of the cosine function. The cosine function has a period of $360^{\circ}$, which means that the cosine function repeats itself every $360^{\circ}$. We can use this property to find all possible solutions.

Finding All Possible Solutions

Using the periodicity of the cosine function, we can find all possible solutions to the equation $\cos x = 0.6663$. We can start by finding the angle whose cosine is 0.6663, which is approximately $44.66^{\circ}$. Then, we can add and subtract multiples of $360^{\circ}$ to find all possible solutions.

Listing All Possible Solutions

Here are all possible solutions to the equation $\cos x = 0.6663$ in the interval $\left[0^{\circ}, 360^{\circ}\right]$:

• $44.66^{\circ}$
• $360^{\circ} - 44.66^{\circ} = 315.34^{\circ}$

Conclusion

In this article, we have found all possible solutions to the equation $2 \cos x - 1 = 0.3326$ in the interval $\left[0^{\circ}, 360^{\circ}\right]$. We have used various trigonometric identities and properties to solve this equation and find all possible solutions. The solutions are $44.66^{\circ}$ and $315.34^{\circ}$. These solutions are valid in the interval $\left[0^{\circ}, 360^{\circ}\right]$.

Final Answer

The final answer is: $\boxed{44.66}$

Introduction

In our previous article, we discussed how to find solutions to the trigonometric equation $2 \cos x - 1 = 0.3326$ in the interval $\left[0^{\circ}, 360^{\circ}\right]$. We used various trigonometric identities and properties to solve this equation and find all possible solutions. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the significance of the interval $\left[0^{\circ}, 360^{\circ}\right]$ in this problem?

A: The interval $\left[0^{\circ}, 360^{\circ}\right]$ is significant because it represents the range of possible values for the angle $x$. The cosine function has a period of $360^{\circ}$, which means that the cosine function repeats itself every $360^{\circ}$. Therefore, we need to find all possible solutions in this interval to ensure that we have considered all possible values of the angle $x$.

Q: How do we use the periodicity of the cosine function to find all possible solutions?

A: We use the periodicity of the cosine function by adding and subtracting multiples of $360^{\circ}$ to the initial solution. This allows us to find all possible solutions in the interval $\left[0^{\circ}, 360^{\circ}\right]$.

Q: What is the relationship between the cosine function and the angle $x$?

A: The cosine function is related to the angle $x$ by the equation $\cos x = 0.6663$. This equation represents the relationship between the cosine function and the angle $x$.

Q: How do we find the angle whose cosine is 0.6663?

A: We can use a calculator or a trigonometric table to find the angle whose cosine is 0.6663. The angle whose cosine is 0.6663 is approximately $44.66^{\circ}$.

Q: What are the possible solutions to the equation $\cos x = 0.6663$ in the interval $\left[0^{\circ}, 360^{\circ}\right]$?

A: The possible solutions to the equation $\cos x = 0.6663$ in the interval $\left[0^{\circ}, 360^{\circ}\right]$ are $44.66^{\circ}$ and $315.34^{\circ}$.

Q: How do we use the inverse cosine function to find the angle whose cosine is 0.6663?

A: We use the inverse cosine function, denoted by $\cos^{-1}$, to find the angle whose cosine is 0.6663. The inverse cosine function returns the angle whose cosine is a given value.

Q: What is the final answer to the equation $2 \cos x - 1 = 0.3326$ in the interval $\left[0^{\circ}, 360^{\circ}\right]$?

A: The final answer to the equation $2 \cos x - 1 = 0.3326$ in the interval $\left[0^{\circ}, 360^{\circ}\right]$ is $44.66^{\circ}$ and $315.34^{\circ}$.

Conclusion

In this article, we have answered some frequently asked questions related to finding solutions to the trigonometric equation $2 \cos x - 1 = 0.3326$ in the interval $\left[0^{\circ}, 360^{\circ}\right]$. We have discussed the significance of the interval, the relationship between the cosine function and the angle $x$, and how to use the periodicity of the cosine function to find all possible solutions.

Final Answer

The final answer is: $\boxed{44.66}$