Solving Cos(x) = 0.7252 Finding X In The Interval [0°, 360°]

In the realm of trigonometry, solving equations is a fundamental skill. This article delves into the process of finding the values of $x$ that satisfy the equation $\cos x = 0.7252$ within the interval $[0^{\circ}, 360^{\circ}]$. We will explore the concepts, techniques, and steps involved in arriving at the solution, providing a comprehensive guide for students and enthusiasts alike.

Understanding the Cosine Function and its Properties

Before we dive into solving the equation, it's crucial to have a firm grasp of the cosine function and its properties. The cosine function, denoted as $\cos x$, is one of the primary trigonometric functions, relating an angle in a right triangle to the ratio of the adjacent side to the hypotenuse. It's a periodic function, meaning its values repeat over regular intervals. The period of the cosine function is $360^{\circ}$, which means that $\cos(x + 360^{\circ}) = \cos x$ for any angle $x$.

The cosine function's graph is a wave-like curve that oscillates between -1 and 1. It starts at a maximum value of 1 at $x = 0^{\circ}$, decreases to 0 at $x = 90^{\circ}$, reaches a minimum value of -1 at $x = 180^{\circ}$, returns to 0 at $x = 270^{\circ}$, and completes its cycle by reaching 1 again at $x = 360^{\circ}$. This cyclical nature is essential to understanding why trigonometric equations often have multiple solutions within a given interval.

The key properties of the cosine function that are relevant to solving equations include:

• Periodicity: $\cos(x + 360^{\circ}k) = \cos x$ for any integer $k$.
• Even function: $\cos(-x) = \cos x$.
• Values in different quadrants: Cosine is positive in the first and fourth quadrants, and negative in the second and third quadrants.

These properties will be instrumental in finding all the solutions to our equation within the specified interval.

Finding the Principal Value

To begin solving the equation $\cos x = 0.7252$, we first need to find the principal value of $x$. The principal value is the angle within the interval $[0^{\circ}, 90^{\circ}]$ (the first quadrant) whose cosine is 0.7252. We can find this value using the inverse cosine function, also known as arccosine, denoted as $\cos^{-1}$ or arccos.

Using a calculator, we can compute:

$x = \cos^{-1}(0.7252) \approx 43.54^{\circ}$

This is our principal value, which lies in the first quadrant. However, since the cosine function is positive in both the first and fourth quadrants, there will be another solution in the fourth quadrant within the interval $[0^{\circ}, 360^{\circ}]$.

Identifying the Quadrants

As mentioned earlier, the cosine function is positive in the first and fourth quadrants. This is a crucial piece of information when solving trigonometric equations. The unit circle provides a visual representation of the trigonometric functions and their signs in different quadrants. In the unit circle:

• The first quadrant (0° to 90°) is where both cosine and sine are positive.
• The second quadrant (90° to 180°) is where sine is positive and cosine is negative.
• The third quadrant (180° to 270°) is where both sine and cosine are negative.
• The fourth quadrant (270° to 360°) is where cosine is positive and sine is negative.

Since we are looking for solutions to $\cos x = 0.7252$, which is a positive value, we know that the solutions must lie in the first and fourth quadrants. We have already found the solution in the first quadrant, which is approximately 43.54°. Now, let's find the solution in the fourth quadrant.

Finding the Solution in the Fourth Quadrant

To find the solution in the fourth quadrant, we utilize the property that the cosine function is positive in the fourth quadrant. The angle in the fourth quadrant that has the same cosine value as our principal value can be found by subtracting the principal value from 360°:

$x = 360^{\circ} - \text{principal value}$

In our case, the principal value is approximately 43.54°, so the fourth quadrant solution is:

$x = 360^{\circ} - 43.54^{\circ} \approx 316.46^{\circ}$

Therefore, the solutions in the interval $[0^{\circ}, 360^{\circ}]$ are approximately 43.54° and 316.46°.

Rounding to the Nearest Degree

The problem asks for the answers to the nearest degree. So, we need to round our solutions to the nearest whole number.

Rounding 43.54° to the nearest degree gives us 44°.

Rounding 316.46° to the nearest degree gives us 316°.

Therefore, the values of $x$ for which $\cos x = 0.7252$ in the interval $[0^{\circ}, 360^{\circ}]$, rounded to the nearest degree, are 44° and 316°.

Verifying the Solutions

It's always a good practice to verify the solutions to ensure they are correct. We can do this by plugging the values back into the original equation and checking if the result is close to 0.7252.

For $x = 44^{\circ}$:

$\cos(44^{\circ}) \approx 0.7193$

This is close to 0.7252, considering we rounded the angle to the nearest degree.

For $x = 316^{\circ}$:

$\cos(316^{\circ}) \approx 0.7193$

This is also close to 0.7252, further confirming our solutions.

While the calculated cosine values are slightly different from 0.7252 due to rounding, they are close enough to confirm the accuracy of our solutions. If higher precision is required, one should use the unrounded values in the verification step.

General Solutions and the Importance of the Interval

It's important to note that the solutions we found are specific to the interval $[0^{\circ}, 360^{\circ}]$. The cosine function is periodic, meaning it repeats its values every 360°. Therefore, there are infinitely many solutions to the equation $\cos x = 0.7252$ if we consider all possible angles.

The general solution for $\cos x = 0.7252$ can be expressed as:

$x = 44^{\circ} + 360^{\circ}k$ or $x = 316^{\circ} + 360^{\circ}k$, where $k$ is any integer.

This means that if we add or subtract multiples of 360° from our solutions, we will still get angles that have a cosine of 0.7252. However, the problem specifically asked for solutions within the interval $[0^{\circ}, 360^{\circ}]$, so we only considered the solutions within that range.

Understanding the concept of general solutions is crucial for solving trigonometric equations in more complex scenarios where specific intervals are not provided.

Conclusion

In this article, we have demonstrated how to solve the trigonometric equation $\cos x = 0.7252$ within the interval $[0^{\circ}, 360^{\circ}]$. We began by understanding the properties of the cosine function, including its periodicity and behavior in different quadrants. We then found the principal value using the inverse cosine function and identified the other solution in the fourth quadrant. Finally, we rounded the solutions to the nearest degree and verified their accuracy.

The key steps involved in solving trigonometric equations like this include:

1. Understanding the properties of the trigonometric function involved.
2. Finding the principal value using inverse trigonometric functions.
3. Identifying the quadrants where solutions exist based on the sign of the trigonometric function.
4. Finding all solutions within the specified interval.
5. Rounding the solutions to the required precision.
6. Verifying the solutions.

By mastering these steps, you can confidently solve a wide range of trigonometric equations and apply these skills to various mathematical and scientific problems. Understanding trigonometric functions and their applications is essential in fields like physics, engineering, and computer graphics. This comprehensive guide should equip you with the knowledge and techniques necessary to tackle similar problems with ease and precision. Remember to always consider the interval, the properties of the trigonometric functions, and the importance of verification to ensure accurate solutions. Practice is key to mastering these concepts, so try solving various trigonometric equations to solidify your understanding. Good luck!