The Equation $\sin \left(25^{\circ}\right)=\frac{9}{c}$ Can Be Used To Find The Length Of $\overline{AB}$. Calculate The Value Of $c$.

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Introduction

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In trigonometry, the sine function is used to relate the angles of a triangle to the ratios of the lengths of its sides. The equation $\sin \left(25^{\circ}\right)=\frac{9}{c}$ is a classic example of how the sine function can be used to find the length of a side in a right-angled triangle. In this article, we will explore how to use this equation to calculate the value of $c$, which represents the length of $\overline{AB}$.

Understanding the Equation

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The equation $\sin \left(25^{\circ}\right)=\frac{9}{c}$ is a trigonometric equation that relates the sine of an angle to the ratio of two sides of a right-angled triangle. In this case, the angle is $25^{\circ}$, and the ratio of the sides is $\frac{9}{c}$. To solve for $c$, we need to isolate $c$ on one side of the equation.

Solving for c

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To solve for $c$, we can start by multiplying both sides of the equation by $c$. This gives us:

$$c \cdot \sin \left(25^{\circ}\right) = 9$$

Next, we can divide both sides of the equation by $\sin \left(25^{\circ}\right)$. This gives us:

$$c = \frac{9}{\sin \left(25^{\circ}\right)}$$

Calculating the Value of c

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Now that we have the equation for $c$, we can calculate its value. To do this, we need to know the value of $\sin \left(25^{\circ}\right)$. We can use a calculator or a trigonometric table to find this value.

Using a calculator, we find that $\sin \left(25^{\circ}\right) \approx 0.4226$. Substituting this value into the equation for $c$, we get:

$$c = \frac{9}{0.4226} \approx 21.32$$

Conclusion

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In this article, we used the equation $\sin \left(25^{\circ}\right)=\frac{9}{c}$ to find the length of $\overline{AB}$. We started by understanding the equation and then solved for $c$ by isolating it on one side of the equation. Finally, we calculated the value of $c$ using a calculator or a trigonometric table.

The Importance of Trigonometry

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Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and computer science. The equation $\sin \left(25^{\circ}\right)=\frac{9}{c}$ is a classic example of how trigonometry can be used to solve real-world problems.

Real-World Applications of Trigonometry

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Trigonometry has numerous real-world applications, including:

• Physics: Trigonometry is used to describe the motion of objects in terms of their position, velocity, and acceleration.
• Engineering: Trigonometry is used to design and build structures such as bridges, buildings, and roads.
• Computer Science: Trigonometry is used in computer graphics to create 3D models and animations.
• Navigation: Trigonometry is used in navigation to determine the position and direction of objects.

Conclusion

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In conclusion, the equation $\sin \left(25^{\circ}\right)=\frac{9}{c}$ is a classic example of how trigonometry can be used to solve real-world problems. By understanding the equation and solving for $c$, we can calculate the length of $\overline{AB}$. Trigonometry is a fundamental subject that has numerous applications in various fields, and its importance cannot be overstated.

Final Thoughts

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The equation $\sin \left(25^{\circ}\right)=\frac{9}{c}$ is a simple yet powerful example of how trigonometry can be used to solve real-world problems. By understanding the equation and solving for $c$, we can calculate the length of $\overline{AB}$. This article has demonstrated the importance of trigonometry and its numerous real-world applications.

References

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• Trigonometry: A branch of mathematics that deals with the relationships between the sides and angles of triangles.
• Sine function: A trigonometric function that relates the angle of a triangle to the ratio of the lengths of its sides.
• Right-angled triangle: A triangle with one angle that is equal to 90 degrees.

Further Reading

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• Trigonometry for Dummies: A comprehensive guide to trigonometry for beginners.
• Trigonometry: A First Course: A textbook on trigonometry for students.
• Trigonometry: A Second Course: A textbook on advanced trigonometry for students.
  

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Introduction

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Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and computer science. In this article, we will answer some of the most frequently asked questions about trigonometry.

Q: What is trigonometry?

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A: Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and computer science.

Q: What are the basic trigonometric functions?

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A: The basic trigonometric functions are:

• Sine (sin): The ratio of the length of the side opposite an angle to the length of the hypotenuse.
• Cosine (cos): The ratio of the length of the side adjacent to an angle to the length of the hypotenuse.
• Tangent (tan): The ratio of the length of the side opposite an angle to the length of the side adjacent to an angle.

Q: What is the sine function?

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A: The sine function is a trigonometric function that relates the angle of a triangle to the ratio of the lengths of its sides. It is defined as the ratio of the length of the side opposite an angle to the length of the hypotenuse.

Q: What is the cosine function?

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A: The cosine function is a trigonometric function that relates the angle of a triangle to the ratio of the lengths of its sides. It is defined as the ratio of the length of the side adjacent to an angle to the length of the hypotenuse.

Q: What is the tangent function?

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A: The tangent function is a trigonometric function that relates the angle of a triangle to the ratio of the lengths of its sides. It is defined as the ratio of the length of the side opposite an angle to the length of the side adjacent to an angle.

Q: What is the Pythagorean theorem?

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A: The Pythagorean theorem is a fundamental concept in trigonometry that states that the square of the length of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the lengths of the other two sides.

Q: How do I use trigonometry in real life?

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A: Trigonometry has numerous applications in various fields, including physics, engineering, and computer science. Some examples of how trigonometry is used in real life include:

• Navigation: Trigonometry is used in navigation to determine the position and direction of objects.
• Physics: Trigonometry is used to describe the motion of objects in terms of their position, velocity, and acceleration.
• Engineering: Trigonometry is used to design and build structures such as bridges, buildings, and roads.
• Computer Science: Trigonometry is used in computer graphics to create 3D models and animations.

Q: What are some common trigonometric identities?

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A: Some common trigonometric identities include:

• sin^2(x) + cos^2(x) = 1
• tan(x) = sin(x) / cos(x)
• cot(x) = cos(x) / sin(x)

Q: How do I solve trigonometric equations?

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A: To solve trigonometric equations, you can use various techniques such as:

• Isolating the trigonometric function: Isolate the trigonometric function on one side of the equation.
• Using trigonometric identities: Use trigonometric identities to simplify the equation.
• Using inverse trigonometric functions: Use inverse trigonometric functions to solve for the angle.

Q: What are some common trigonometric formulas?

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A: Some common trigonometric formulas include:

• sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
• cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
• tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b))

Conclusion

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In conclusion, trigonometry is a fundamental subject that has numerous applications in various fields. By understanding the basic trigonometric functions, identities, and formulas, you can solve a wide range of problems in physics, engineering, and computer science. We hope that this article has provided you with a comprehensive overview of trigonometry and its applications.

References

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• Trigonometry: A branch of mathematics that deals with the relationships between the sides and angles of triangles.
• Sine function: A trigonometric function that relates the angle of a triangle to the ratio of the lengths of its sides.
• Cosine function: A trigonometric function that relates the angle of a triangle to the ratio of the lengths of its sides.
• Tangent function: A trigonometric function that relates the angle of a triangle to the ratio of the lengths of its sides.

Further Reading

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• Trigonometry for Dummies: A comprehensive guide to trigonometry for beginners.
• Trigonometry: A First Course: A textbook on trigonometry for students.
• Trigonometry: A Second Course: A textbook on advanced trigonometry for students.