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 used to relate the angles of a triangle to the ratios of the lengths of its sides. 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 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 triangle. In this case, the angle is $40^{\circ}$, and the ratio is $\frac{b}{20}$, where $b$ is the length of the line segment AC.

To solve for $b$, we need to isolate it on one side of the equation. We can do this by multiplying both sides of the equation by 20, which gives us:

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

Using a Calculator to Find the Value of $\sin \left(40^{\circ}\right)$

To find the value of $\sin \left(40^{\circ}\right)$, we can use a calculator. The sine function is typically found on the calculator as the "sin" button. We can enter the angle $40^{\circ}$ and press the "sin" button to get the value of $\sin \left(40^{\circ}\right)$.

Using a calculator, we find that $\sin \left(40^{\circ}\right) \approx 0.6428$.

Substituting the Value of $\sin \left(40^{\circ}\right)$ into the Equation

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

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

$$20 \times 0.6428 = b$$

Solving for $b$

To solve for $b$, we can multiply 20 by 0.6428:

$$b = 20 \times 0.6428$$

$$b \approx 12.856$$

Rounding to the Nearest Hundredth

Since the answer choices are given to the nearest hundredth, we can round our answer to the nearest hundredth:

$$b \approx 12.86$$

However, we are given answer choices in the form of decimal numbers. We can convert our answer to a decimal number by dividing by 1.

$$b \approx 12.86$$

Comparing Our Answer to the Answer Choices

Now that we have our answer, we can compare it to the answer choices:

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

Our answer, 12.86 cm, is closest to answer choice B, 12.9 cm.

Conclusion

In this article, we used the equation $\sin \left(40^{\circ}\right) = \frac{b}{20}$ to solve for the length of line segment AC. We found that the length of line segment AC is approximately 12.86 cm, which is closest to answer choice B, 12.9 cm.

Final Answer

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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 its solution.

Q&A

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

A: The sine function is a trigonometric function that relates the angles of a triangle to the ratios of the lengths of its sides. In the equation $\sin \left(40^{\circ}\right) = \frac{b}{20}$, the sine function is used to relate the angle $40^{\circ}$ to the ratio of the length of line segment AC to 20.

Q: How do I use a calculator to find the value of $\sin \left(40^{\circ}\right)$?

A: To find the value of $\sin \left(40^{\circ}\right)$, you can use a calculator and enter the angle $40^{\circ}$ and press the "sin" button. The calculator will then display the value of $\sin \left(40^{\circ}\right)$.

Q: Why do we need to multiply both sides of the equation by 20?

A: We need to multiply both sides of the equation by 20 to isolate the variable $b$ on one side of the equation. This allows us to solve for the value of $b$.

Q: Can I use a different method to solve for the value of $b$?

A: Yes, you can use a different method to solve for the value of $b$. For example, you can use the inverse sine function to find the value of $b$. However, the method we used in this article is a straightforward and efficient way to solve for the value of $b$.

Q: What if I get a different answer when I use a different method to solve for the value of $b$?

A: If you get a different answer when you use a different method to solve for the value of $b$, it may be due to a mistake in your calculation or a misunderstanding of the equation. Make sure to double-check your work and ensure that you are using the correct method to solve for the value of $b$.

Q: Can I use this equation to solve for the length of line segment AC in other triangles?

A: Yes, you can use this equation to solve for the length of line segment AC in other triangles. However, you will need to ensure that the triangle is a right triangle and that the angle is $40^{\circ}$.

Q: What if the angle is not $40^{\circ}$?

A: If the angle is not $40^{\circ}$, you will need to use a different equation to solve for the length of line segment AC. The equation we used in this article is specific to the angle $40^{\circ}$.

Conclusion

In this article, we answered some frequently asked questions related to the equation $\sin \left(40^{\circ}\right) = \frac{b}{20}$ and its solution. We hope that this article has been helpful in clarifying any confusion and providing additional information on the topic.

Final Answer

The final answer is: $\boxed{12.9}$