For What Value Of X X X Is Sin ⁡ ( X ) = Cos ⁡ ( 32 ∘ \sin(x) = \cos(32^{\circ} Sin ( X ) = Cos ( 3 2 ∘ ], Where 0 ∘ \textless X \textless 90 ∘ 0^{\circ} \ \textless \ X \ \textless \ 90^{\circ} 0 ∘ \textless X \textless 9 0 ∘ ?A. 32 ∘ 32^{\circ} 3 2 ∘ B. 64 ∘ 64^{\circ} 6 4 ∘ C. 13 ∘ 13^{\circ} 1 3 ∘ D. 58 ∘ 58^{\circ} 5 8 ∘

Introduction

Trigonometric equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties and relationships between different trigonometric functions. In this article, we will focus on solving the equation $\sin(x) = \cos(32^{\circ})$, where $0^{\circ} \ \textless \ x \ \textless \ 90^{\circ}$. This equation involves the sine and cosine functions, which are two of the most important trigonometric functions. We will use various trigonometric identities and properties to solve this equation and find the value of $x$.

Understanding the Sine and Cosine Functions

Before we dive into solving the equation, let's briefly review the sine and cosine functions. The sine function, denoted by $\sin(x)$, is a periodic function that oscillates between $-1$ and $1$. The cosine function, denoted by $\cos(x)$, is also a periodic function that oscillates between $-1$ and $1$. The sine and cosine functions are related to each other through the Pythagorean identity: $\sin^2(x) + \cos^2(x) = 1$.

The Equation $\sin(x) = \cos(32^{\circ})$

Now, let's focus on the equation $\sin(x) = \cos(32^{\circ})$. We are given that $0^{\circ} \ \textless \ x \ \textless \ 90^{\circ}$, which means that $x$ is an acute angle. Since the sine and cosine functions are periodic, we can use the properties of these functions to simplify the equation.

Using the Co-function Identity

One of the most important properties of the sine and cosine functions is the co-function identity: $\sin(x) = \cos(90^{\circ} - x)$. This identity states that the sine of an angle is equal to the cosine of its complementary angle. We can use this identity to rewrite the equation $\sin(x) = \cos(32^{\circ})$ as $\cos(90^{\circ} - x) = \cos(32^{\circ})$.

Solving for $x$

Now that we have rewritten the equation, we can solve for $x$. Since the cosine function is periodic, we can add or subtract multiples of $360^{\circ}$ to the angle without changing the value of the cosine function. Therefore, we can write $90^{\circ} - x = 32^{\circ} + 360^{\circ}n$, where $n$ is an integer.

Finding the Value of $x$

To find the value of $x$, we can isolate $x$ in the equation $90^{\circ} - x = 32^{\circ} + 360^{\circ}n$. Subtracting $32^{\circ}$ from both sides, we get $58^{\circ} - x = 360^{\circ}n$. Adding $x$ to both sides, we get $58^{\circ} = x + 360^{\circ}n$. Subtracting $360^{\circ}n$ from both sides, we get $x = 58^{\circ} - 360^{\circ}n$.

Finding the Value of $n$

Since $0^{\circ} \ \textless \ x \ \textless \ 90^{\circ}$, we know that $x$ is an acute angle. Therefore, we can find the value of $n$ by setting $x = 58^{\circ} - 360^{\circ}n$ and solving for $n$. Since $x$ is an acute angle, we know that $x \ \textless \ 90^{\circ}$. Therefore, we can set $58^{\circ} - 360^{\circ}n \ \textless \ 90^{\circ}$ and solve for $n$.

Solving for $n$

To solve for $n$, we can add $360^{\circ}n$ to both sides of the inequality $58^{\circ} - 360^{\circ}n \ \textless \ 90^{\circ}$. This gives us $58^{\circ} \ \textless \ 90^{\circ} + 360^{\circ}n$. Subtracting $58^{\circ}$ from both sides, we get $0^{\circ} \ \textless \ 32^{\circ} + 360^{\circ}n$. Subtracting $32^{\circ}$ from both sides, we get $-32^{\circ} \ \textless \ 360^{\circ}n$. Dividing both sides by $360^{\circ}$, we get $-\frac{32^{\circ}}{360^{\circ}} \ \textless \ n$. Simplifying, we get $-\frac{1}{11.25} \ \textless \ n$.

Finding the Value of $x$

Now that we have found the value of $n$, we can find the value of $x$. Substituting $n = 0$ into the equation $x = 58^{\circ} - 360^{\circ}n$, we get $x = 58^{\circ} - 360^{\circ}(0)$. Simplifying, we get $x = 58^{\circ}$.

Conclusion

In this article, we have solved the equation $\sin(x) = \cos(32^{\circ})$, where $0^{\circ} \ \textless \ x \ \textless \ 90^{\circ}$. We have used various trigonometric identities and properties to simplify the equation and find the value of $x$. The final answer is $x = 58^{\circ}$.

Final Answer

The final answer is $\boxed{58^{\circ}}$.

Introduction

In our previous article, we solved the equation $\sin(x) = \cos(32^{\circ})$, where $0^{\circ} \ \textless \ x \ \textless \ 90^{\circ}$. We used various trigonometric identities and properties to simplify the equation and find the value of $x$. In this article, we will answer some common questions related to solving trigonometric equations.

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

A: The sine and cosine functions are two of the most important trigonometric functions. The sine function, denoted by $\sin(x)$, is a periodic function that oscillates between $-1$ and $1$. The cosine function, denoted by $\cos(x)$, is also a periodic function that oscillates between $-1$ and $1$. The sine and cosine functions are related to each other through the Pythagorean identity: $\sin^2(x) + \cos^2(x) = 1$.

Q: How do I solve a trigonometric equation?

A: To solve a trigonometric equation, you need to use various trigonometric identities and properties. You can start by simplifying the equation using the Pythagorean identity, and then use the co-function identity to rewrite the equation in terms of a single trigonometric function. Finally, you can use the inverse trigonometric functions to find the value of the variable.

Q: What is the co-function identity?

A: The co-function identity states that the sine of an angle is equal to the cosine of its complementary angle. Mathematically, this can be written as $\sin(x) = \cos(90^{\circ} - x)$. This identity is useful for simplifying trigonometric equations and finding the value of the variable.

Q: How do I use the inverse trigonometric functions?

A: The inverse trigonometric functions are used to find the value of the variable in a trigonometric equation. For example, if you have the equation $\sin(x) = \cos(32^{\circ})$, you can use the inverse sine function to find the value of $x$. The inverse sine function is denoted by $\sin^{-1}(x)$.

Q: What is the difference between the sine and cosine functions in terms of their graphs?

A: The sine and cosine functions have similar graphs, but they are shifted by $90^{\circ}$. The sine function has a maximum value of $1$ and a minimum value of $-1$, while the cosine function has a maximum value of $1$ and a minimum value of $-1$. The graphs of the sine and cosine functions are also periodic, with a period of $360^{\circ}$.

Q: How do I find the value of the variable in a trigonometric equation?

A: To find the value of the variable in a trigonometric equation, you need to use the inverse trigonometric functions. For example, if you have the equation $\sin(x) = \cos(32^{\circ})$, you can use the inverse sine function to find the value of $x$. The inverse sine function is denoted by $\sin^{-1}(x)$.

Q: What is the significance of the Pythagorean identity?

A: The Pythagorean identity is a fundamental identity in trigonometry that relates the sine and cosine functions. It states that $\sin^2(x) + \cos^2(x) = 1$, which is a fundamental property of the sine and cosine functions. The Pythagorean identity is used to simplify trigonometric equations and find the value of the variable.

Conclusion

In this article, we have answered some common questions related to solving trigonometric equations. We have discussed the difference between the sine and cosine functions, how to solve a trigonometric equation, the co-function identity, and how to use the inverse trigonometric functions. We have also discussed the significance of the Pythagorean identity and how it is used to simplify trigonometric equations and find the value of the variable.

Final Answer

The final answer is $\boxed{58^{\circ}}$.