For What Value Of $x$ Is $\cos(x) = \sin(14^{\circ})$, Where $ 0 ∘ \textless X \textless 90 ∘ 0^{\circ} \ \textless \ X \ \textless \ 90^{\circ} 0 ∘ \textless X \textless 9 0 ∘ [/tex]?A. $14^{\circ}$ B. $76^{\circ}$ C.

Introduction

Trigonometric equations are a fundamental concept in mathematics, and solving them requires a deep understanding of trigonometric functions and their properties. In this article, we will focus on solving the equation $\cos(x) = \sin(14^{\circ})$, where $0^{\circ} < x < 90^{\circ}$. This equation involves the cosine and sine functions, which are two of the most important trigonometric functions.

Understanding Trigonometric Functions

Before we dive into solving the equation, let's briefly review the trigonometric functions involved. The cosine function, denoted by $\cos(x)$, is a periodic function that represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle. The sine function, denoted by $\sin(x)$, is also a periodic function that represents the ratio of the opposite side to the hypotenuse in a right-angled triangle.

Properties of Trigonometric Functions

To solve the equation, we need to use the properties of trigonometric functions. One of the most important properties is the co-function identity, which states that $\cos(x) = \sin(90^{\circ} - x)$. This identity allows us to rewrite the equation in terms of the sine function.

Rewriting the Equation

Using the co-function identity, we can rewrite the equation as follows:

$$\cos(x) = \sin(14^{\circ})$$

$$\sin(90^{\circ} - x) = \sin(14^{\circ})$$

Equating Angles

Since the sine function is periodic, we can equate the angles inside the sine function:

$$90^{\circ} - x = 14^{\circ}$$

Solving for x

Now, we can solve for x by isolating it on one side of the equation:

$$x = 90^{\circ} - 14^{\circ}$$

$$x = 76^{\circ}$$

Conclusion

In this article, we solved the equation $\cos(x) = \sin(14^{\circ})$, where $0^{\circ} < x < 90^{\circ}$. We used the co-function identity to rewrite the equation in terms of the sine function and then equated the angles inside the sine function. Finally, we solved for x by isolating it on one side of the equation. The solution to the equation is $x = 76^{\circ}$.

Final Answer

The final answer to the equation $\cos(x) = \sin(14^{\circ})$, where $0^{\circ} < x < 90^{\circ}$, is $x = 76^{\circ}$.

Additional Tips and Tricks

• When solving trigonometric equations, it's essential to use the co-function identity to rewrite the equation in terms of the sine or cosine function.
• Equating angles inside the sine or cosine function can help simplify the equation and solve for x.
• Remember to check the domain of the equation to ensure that the solution is valid.

Common Mistakes to Avoid

• Failing to use the co-function identity to rewrite the equation in terms of the sine or cosine function.
• Not equating angles inside the sine or cosine function.
• Not checking the domain of the equation to ensure that the solution is valid.

Real-World Applications

Trigonometric equations have numerous real-world applications, including:

• Physics: Trigonometric equations are used to describe the motion of objects in terms of position, velocity, and acceleration.
• Engineering: Trigonometric equations are used to design and analyze electrical circuits, mechanical systems, and other engineering applications.
• Computer Science: Trigonometric equations are used in computer graphics, game development, and other areas of computer science.

Conclusion

In conclusion, solving trigonometric equations requires a deep understanding of trigonometric functions and their properties. By using the co-function identity and equating angles inside the sine or cosine function, we can solve equations like $\cos(x) = \sin(14^{\circ})$, where $0^{\circ} < x < 90^{\circ}$. The solution to the equation is $x = 76^{\circ}$.

Q: What is the co-function identity?

A: The co-function identity is a property of trigonometric functions that states $\cos(x) = \sin(90^{\circ} - x)$ and $\sin(x) = \cos(90^{\circ} - x)$. This identity allows us to rewrite the equation in terms of the sine or cosine function.

Q: How do I use the co-function identity to solve a trigonometric equation?

A: To use the co-function identity, rewrite the equation in terms of the sine or cosine function by replacing the cosine function with the sine function and vice versa. Then, equate the angles inside the sine or cosine function to solve for x.

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

A: The sine function represents the ratio of the opposite side to the hypotenuse in a right-angled triangle, while the cosine function represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle.

Q: How do I know which trigonometric function to use when solving an equation?

A: When solving a trigonometric equation, use the co-function identity to rewrite the equation in terms of the sine or cosine function. Then, equate the angles inside the sine or cosine function to solve for x.

Q: What is the domain of a trigonometric function?

A: The domain of a trigonometric function is the set of all possible input values (angles) for which the function is defined. For the sine and cosine functions, the domain is all real numbers.

Q: How do I check the domain of a trigonometric equation?

A: To check the domain of a trigonometric equation, ensure that the input values (angles) are within the domain of the sine or cosine function. For the sine and cosine functions, the domain is all real numbers.

Q: What are some common mistakes to avoid when solving trigonometric equations?

A: Some common mistakes to avoid when solving trigonometric equations include:

• Failing to use the co-function identity to rewrite the equation in terms of the sine or cosine function.
• Not equating angles inside the sine or cosine function.
• Not checking the domain of the equation to ensure that the solution is valid.

Q: What are some real-world applications of trigonometric equations?

A: Trigonometric equations have numerous real-world applications, including:

• Physics: Trigonometric equations are used to describe the motion of objects in terms of position, velocity, and acceleration.
• Engineering: Trigonometric equations are used to design and analyze electrical circuits, mechanical systems, and other engineering applications.
• Computer Science: Trigonometric equations are used in computer graphics, game development, and other areas of computer science.

Q: How do I practice solving trigonometric equations?

A: To practice solving trigonometric equations, try the following:

• Start with simple equations and gradually move on to more complex ones.
• Use online resources, such as trigonometric equation solvers or practice problems.
• Work with a partner or join a study group to practice solving trigonometric equations together.

Q: What are some additional tips and tricks for solving trigonometric equations?

A: Some additional tips and tricks for solving trigonometric equations include:

• Using the co-function identity to rewrite the equation in terms of the sine or cosine function.
• Equating angles inside the sine or cosine function to solve for x.
• Checking the domain of the equation to ensure that the solution is valid.

Conclusion

In conclusion, solving trigonometric equations requires a deep understanding of trigonometric functions and their properties. By using the co-function identity and equating angles inside the sine or cosine function, we can solve equations like $\cos(x) = \sin(14^{\circ})$, where $0^{\circ} < x < 90^{\circ}$. The solution to the equation is $x = 76^{\circ}$.