The Equation $\sin \left(40^{\circ}\right) = \frac{b}{20}$ Can Be Used To Determine The Length Of Line Segment AC. A. 11.8 Cm B. 12.9 Cm C. 14.9 Cm D. 15.3 Cm

Introduction

In trigonometry, the sine function is a fundamental concept used to describe the relationship between the angles and side lengths of triangles. The equation $\sin \left(40^{\circ}\right) = \frac{b}{20}$ is a classic example of how the sine function can be used to solve for the length of a line segment in a right-angled triangle. In this article, we will explore the solution to this equation and determine the length of line segment AC.

Understanding the Equation

The equation $\sin \left(40^{\circ}\right) = \frac{b}{20}$ involves the sine function, which is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right-angled triangle. In this case, the angle is $40^{\circ}$, and the ratio is equal to $\frac{b}{20}$, where $b$ is the length of the side opposite the angle.

Solving for b

To solve for $b$, we can start by isolating the sine function on one side of the equation. This can be done by multiplying both sides of the equation by 20, which gives us:

$$20 \sin \left(40^{\circ}\right) = b$$

Next, we can use a calculator to find the value of $\sin \left(40^{\circ}\right)$. The sine function is typically evaluated using a calculator or a trigonometric table. In this case, we can use a calculator to find that $\sin \left(40^{\circ}\right) \approx 0.643$.

Substituting the Value of Sin(40°)

Now that we have the value of $\sin \left(40^{\circ}\right)$, we can substitute it into the equation:

$$20 \sin \left(40^{\circ}\right) = b$$

$$20 \times 0.643 = b$$

$$12.86 = b$$

Rounding the Answer

The value of $b$ is approximately 12.86 cm. However, we need to round this value to the nearest tenth of a centimeter, as the answer choices are given in this format. Therefore, we can round 12.86 cm to 12.9 cm.

Conclusion

In conclusion, the equation $\sin \left(40^{\circ}\right) = \frac{b}{20}$ can be used to determine the length of line segment AC. By solving for $b$, we found that the length of line segment AC is approximately 12.9 cm.

Comparison with Answer Choices

Now that we have the solution to the equation, we can compare it with the answer choices:

A. 11.8 cm B. 12.9 cm C. 14.9 cm D. 15.3 cm

Based on our solution, we can see that the correct answer is B. 12.9 cm.

Final Thoughts

In this article, we explored the solution to the equation $\sin \left(40^{\circ}\right) = \frac{b}{20}$ and determined the length of line segment AC. We used the sine function to solve for $b$ and found that the length of line segment AC is approximately 12.9 cm. This problem is a classic example of how the sine function can be used to solve for the length of a line segment in a right-angled triangle.

Introduction

In our previous article, we explored the solution to the equation $\sin \left(40^{\circ}\right) = \frac{b}{20}$ and determined the length of line segment AC. In this article, we will answer some frequently asked questions related to the equation and provide additional insights into the solution.

Q&A

Q: What is the sine function and how is it used in the equation?

A: The sine function is a fundamental concept in trigonometry that describes the relationship between the angles and side lengths of triangles. In the equation $\sin \left(40^{\circ}\right) = \frac{b}{20}$, the sine function is used to describe the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right-angled triangle.

Q: How do I evaluate the sine function on a calculator?

A: To evaluate the sine function on a calculator, you can use the following steps:

1. Enter the angle in degrees (in this case, 40°).
2. Press the sine button (usually labeled as "sin" or "sin(x)").
3. The calculator will display the value of the sine function.

Q: What is the value of the sine function for 40°?

A: The value of the sine function for 40° is approximately 0.643.

Q: How do I solve for b in the equation?

A: To solve for b, you can multiply both sides of the equation by 20, which gives you:

$$20 \sin \left(40^{\circ}\right) = b$$

Then, you can substitute the value of the sine function (approximately 0.643) into the equation and solve for b.

Q: What is the length of line segment AC?

A: The length of line segment AC is approximately 12.9 cm.

Q: How do I round the answer to the nearest tenth of a centimeter?

A: To round the answer to the nearest tenth of a centimeter, you can look at the hundredth place value (in this case, 0.86). If the hundredth place value is 5 or greater, you round up. If the hundredth place value is less than 5, you round down. In this case, you would round 12.86 cm to 12.9 cm.

Q: What are the answer choices for the problem?

A: The answer choices for the problem are:

A. 11.8 cm B. 12.9 cm C. 14.9 cm D. 15.3 cm

Q: Which answer choice is correct?

A: The correct answer choice is B. 12.9 cm.

Conclusion

In this article, we answered some frequently asked questions related to the equation $\sin \left(40^{\circ}\right) = \frac{b}{20}$ and provided additional insights into the solution. We hope that this Q&A article has been helpful in clarifying any doubts you may have had about the problem.

Additional Resources

If you are interested in learning more about trigonometry and the sine function, we recommend checking out the following resources:

• Khan Academy: Trigonometry
• Mathway: Sine Function
• Wolfram Alpha: Sine Function

We hope that you find these resources helpful in your studies!