If $5 \cot \theta = 7$, Then Find The Value Of $\frac{7 \sin \theta + 5 \cos \theta}{5 \sin \theta + 7 \cos \theta}$.

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Introduction

In this article, we will explore a trigonometric problem involving the cotangent function and its relationship with the sine and cosine functions. We will use the given equation $5 \cot \theta = 7$ to find the value of the expression $\frac{7 \sin \theta + 5 \cos \theta}{5 \sin \theta + 7 \cos \theta}$. This problem requires a deep understanding of trigonometric identities and their applications.

Understanding the Cotangent Function

The cotangent function is defined as the ratio of the adjacent side to the opposite side in a right-angled triangle. It is the reciprocal of the tangent function, which is the ratio of the opposite side to the adjacent side. The cotangent function can be expressed in terms of the sine and cosine functions as $\cot \theta = \frac{\cos \theta}{\sin \theta}$.

Given Equation

We are given the equation $5 \cot \theta = 7$. We can rewrite this equation using the definition of the cotangent function as $5 \frac{\cos \theta}{\sin \theta} = 7$. This equation can be further simplified to $\frac{5 \cos \theta}{\sin \theta} = 7$.

Relationship Between Sine and Cosine Functions

To find the value of the expression $\frac{7 \sin \theta + 5 \cos \theta}{5 \sin \theta + 7 \cos \theta}$, we need to establish a relationship between the sine and cosine functions. We can use the given equation $\frac{5 \cos \theta}{\sin \theta} = 7$ to express the cosine function in terms of the sine function.

Expressing Cosine in Terms of Sine

We can rearrange the given equation to express the cosine function in terms of the sine function as $\cos \theta = \frac{7 \sin \theta}{5}$. This expression allows us to substitute the cosine function in terms of the sine function in the given expression.

Substituting Cosine in Terms of Sine

We can substitute the expression for the cosine function in terms of the sine function into the given expression as $\frac{7 \sin \theta + 5 \left(\frac{7 \sin \theta}{5}\right)}{5 \sin \theta + 7 \left(\frac{7 \sin \theta}{5}\right)}$. This substitution simplifies the expression and allows us to find its value.

Simplifying the Expression

We can simplify the expression by combining like terms as $\frac{7 \sin \theta + \frac{7 \sin \theta}{1}}{5 \sin \theta + \frac{49 \sin \theta}{5}}$. This simplification allows us to further reduce the expression and find its value.

Further Simplification

We can further simplify the expression by combining like terms as $\frac{\frac{7 \sin \theta + 7 \sin \theta}{1}}{\frac{5 \sin \theta + 49 \sin \theta}{5}}$. This simplification allows us to express the numerator and denominator in terms of a common denominator.

Common Denominator

We can express the numerator and denominator in terms of a common denominator as $\frac{\frac{14 \sin \theta}{1}}{\frac{54 \sin \theta}{5}}$. This expression allows us to simplify the fraction and find its value.

Simplifying the Fraction

We can simplify the fraction by dividing the numerator by the denominator as $\frac{14 \sin \theta}{1} \div \frac{54 \sin \theta}{5}$. This simplification allows us to find the value of the expression.

Final Value

We can simplify the fraction by dividing the numerator by the denominator as $\frac{14}{54} \div \frac{1}{5}$. This simplification allows us to find the final value of the expression.

Conclusion

In this article, we used the given equation $5 \cot \theta = 7$ to find the value of the expression $\frac{7 \sin \theta + 5 \cos \theta}{5 \sin \theta + 7 \cos \theta}$. We established a relationship between the sine and cosine functions, expressed the cosine function in terms of the sine function, and simplified the expression to find its value. The final value of the expression is $\frac{7}{5}$.

Final Answer

The final answer is $\boxed{\frac{7}{5}}$.

References

  • [1] Trigonometry, by Michael Corral
  • [2] Calculus, by Michael Spivak
  • [3] Mathematics, by G.H. Hardy

Keywords

  • Trigonometry
  • Cotangent function
  • Sine function
  • Cosine function
  • Trigonometric identities
  • Right-angled triangle
  • Reciprocal function
  • Tangent function
  • Trigonometric ratios
  • Trigonometric functions
  • Mathematics
  • Calculus
  • Algebra

Q: What is the cotangent function and how is it related to the sine and cosine functions?

A: The cotangent function is defined as the ratio of the adjacent side to the opposite side in a right-angled triangle. It is the reciprocal of the tangent function, which is the ratio of the opposite side to the adjacent side. The cotangent function can be expressed in terms of the sine and cosine functions as $\cot \theta = \frac{\cos \theta}{\sin \theta}$.

Q: How is the given equation $5 \cot \theta = 7$ related to the sine and cosine functions?

A: We can rewrite the given equation using the definition of the cotangent function as $5 \frac{\cos \theta}{\sin \theta} = 7$. This equation can be further simplified to $\frac{5 \cos \theta}{\sin \theta} = 7$.

Q: How can we express the cosine function in terms of the sine function using the given equation?

A: We can rearrange the given equation to express the cosine function in terms of the sine function as $\cos \theta = \frac{7 \sin \theta}{5}$. This expression allows us to substitute the cosine function in terms of the sine function in the given expression.

Q: How can we simplify the expression $\frac{7 \sin \theta + 5 \cos \theta}{5 \sin \theta + 7 \cos \theta}$ using the expression for the cosine function in terms of the sine function?

A: We can substitute the expression for the cosine function in terms of the sine function into the given expression as $\frac{7 \sin \theta + 5 \left(\frac{7 \sin \theta}{5}\right)}{5 \sin \theta + 7 \left(\frac{7 \sin \theta}{5}\right)}$. This substitution simplifies the expression and allows us to find its value.

Q: What is the final value of the expression $\frac{7 \sin \theta + 5 \cos \theta}{5 \sin \theta + 7 \cos \theta}$?

A: The final value of the expression is $\frac{7}{5}$.

Q: What are some common trigonometric identities that are used in solving problems like this?

A: Some common trigonometric identities that are used in solving problems like this include the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$, the reciprocal identity $\cot \theta = \frac{\cos \theta}{\sin \theta}$, and the sum and difference identities for sine and cosine.

Q: How can we use the given equation $5 \cot \theta = 7$ to find the value of the expression $\frac{7 \sin \theta + 5 \cos \theta}{5 \sin \theta + 7 \cos \theta}$?

A: We can use the given equation to express the cosine function in terms of the sine function, and then substitute this expression into the given expression. This allows us to simplify the expression and find its value.

Q: What is the significance of the cotangent function in trigonometry?

A: The cotangent function is an important trigonometric function that is used to describe the relationship between the adjacent side and the opposite side in a right-angled triangle. It is the reciprocal of the tangent function, and is used in a variety of mathematical and scientific applications.

Q: How can we use the expression $\frac{7 \sin \theta + 5 \cos \theta}{5 \sin \theta + 7 \cos \theta}$ in real-world applications?

A: The expression $\frac{7 \sin \theta + 5 \cos \theta}{5 \sin \theta + 7 \cos \theta}$ can be used in a variety of real-world applications, including navigation, engineering, and physics. It can be used to describe the relationship between the sine and cosine functions in a right-angled triangle, and can be used to solve problems involving trigonometry.

Q: What are some common mistakes that people make when solving problems like this?

A: Some common mistakes that people make when solving problems like this include not using the correct trigonometric identities, not simplifying the expression correctly, and not checking the final answer. It is also important to make sure that the given equation is used correctly, and that the expression is simplified in a way that is consistent with the given equation.

Q: How can we use the given equation $5 \cot \theta = 7$ to find the value of the expression $\frac{7 \sin \theta + 5 \cos \theta}{5 \sin \theta + 7 \cos \theta}$ in a more efficient way?

A: We can use the given equation to express the cosine function in terms of the sine function, and then substitute this expression into the given expression. This allows us to simplify the expression and find its value in a more efficient way.

Q: What are some common tools and techniques that are used to solve problems like this?

A: Some common tools and techniques that are used to solve problems like this include the use of trigonometric identities, the use of algebraic manipulations, and the use of geometric reasoning. It is also important to use a calculator or computer program to check the final answer and to make sure that the expression is simplified correctly.

Q: How can we use the expression $\frac{7 \sin \theta + 5 \cos \theta}{5 \sin \theta + 7 \cos \theta}$ to solve problems involving trigonometry?

A: The expression $\frac{7 \sin \theta + 5 \cos \theta}{5 \sin \theta + 7 \cos \theta}$ can be used to solve problems involving trigonometry, including problems involving right-angled triangles, navigation, and engineering. It can be used to describe the relationship between the sine and cosine functions in a right-angled triangle, and can be used to solve problems involving trigonometry.

Q: What are some common applications of the cotangent function in real-world problems?

A: The cotangent function has a variety of applications in real-world problems, including navigation, engineering, and physics. It can be used to describe the relationship between the adjacent side and the opposite side in a right-angled triangle, and can be used to solve problems involving trigonometry.

Q: How can we use the given equation $5 \cot \theta = 7$ to find the value of the expression $\frac{7 \sin \theta + 5 \cos \theta}{5 \sin \theta + 7 \cos \theta}$ in a more general way?

A: We can use the given equation to express the cosine function in terms of the sine function, and then substitute this expression into the given expression. This allows us to simplify the expression and find its value in a more general way.

Q: What are some common mistakes that people make when using the cotangent function in real-world problems?

A: Some common mistakes that people make when using the cotangent function in real-world problems include not using the correct trigonometric identities, not simplifying the expression correctly, and not checking the final answer. It is also important to make sure that the given equation is used correctly, and that the expression is simplified in a way that is consistent with the given equation.

Q: How can we use the expression $\frac{7 \sin \theta + 5 \cos \theta}{5 \sin \theta + 7 \cos \theta}$ to solve problems involving trigonometry in a more efficient way?

A: The expression $\frac{7 \sin \theta + 5 \cos \theta}{5 \sin \theta + 7 \cos \theta}$ can be used to solve problems involving trigonometry in a more efficient way by using the given equation to express the cosine function in terms of the sine function, and then substituting this expression into the given expression.

Q: What are some common tools and techniques that are used to solve problems involving trigonometry?

A: Some common tools and techniques that are used to solve problems involving trigonometry include the use of trigonometric identities, the use of algebraic manipulations, and the use of geometric reasoning. It is also important to use a calculator or computer program to check the final answer and to make sure that the expression is simplified correctly.

Q: How can we use the given equation $5 \cot \theta = 7$ to find the value of the expression $\frac{7 \sin \theta + 5 \cos \theta}{5 \sin \theta + 7 \cos \theta}$ in a more general way?

A: We can use the given equation to express the cosine function in terms of the sine function, and then substitute this expression into the given expression. This allows us to simplify the expression and find its value in a more general way.

Q: What are some common applications of the expression $\frac{7 \sin \theta + 5 \cos \theta}{5 \sin \theta + 7 \cos \theta}$ in real-world problems?

A: The expression $\frac{7 \sin \theta + 5 \cos \theta}{5 \sin \theta + 7 \cos