A Student Uses The Equation Tan ⁡ Θ = S 2 49 \tan \theta = \frac{s^2}{49} Tan Θ = 49 S 2 ​ To Represent The Speed, S S S , In Feet Per Second, Of A Toy Car Driving Around A Circular Track Having An Angle Of Incline Θ \theta Θ , Where $\sin \theta =

Introduction

In mathematics, trigonometric equations are used to represent various real-world problems, such as the motion of objects in physics. A student is given the equation $\tan \theta = \frac{s^2}{49}$, where $\sin \theta = \frac{3}{5}$, to find the speed, $s$, in feet per second, of a toy car driving around a circular track having an angle of incline $\theta$. In this article, we will guide the student through the process of solving this trigonometric equation and provide a step-by-step solution.

Understanding the Given Equation

The given equation is $\tan \theta = \frac{s^2}{49}$, where $\sin \theta = \frac{3}{5}$. To solve for $s$, we need to first find the value of $\tan \theta$. Since we are given the value of $\sin \theta$, we can use the Pythagorean identity to find the value of $\cos \theta$.

Using the Pythagorean Identity

The Pythagorean identity states that $\sin^2 \theta + \cos^2 \theta = 1$. We can rearrange this equation to solve for $\cos \theta$:

$$\cos^2 \theta = 1 - \sin^2 \theta$$

Substituting the given value of $\sin \theta = \frac{3}{5}$, we get:

$$\cos^2 \theta = 1 - \left(\frac{3}{5}\right)^2$$

Simplifying the equation, we get:

$$\cos^2 \theta = 1 - \frac{9}{25}$$

$$\cos^2 \theta = \frac{16}{25}$$

Taking the square root of both sides, we get:

$$\cos \theta = \pm \sqrt{\frac{16}{25}}$$

Since the angle of incline $\theta$ is positive, we take the positive square root:

$$\cos \theta = \frac{4}{5}$$

Finding the Value of $\tan \theta$

Now that we have the values of $\sin \theta$ and $\cos \theta$, we can find the value of $\tan \theta$ using the definition of the tangent function:

$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$

Substituting the values of $\sin \theta = \frac{3}{5}$ and $\cos \theta = \frac{4}{5}$, we get:

$$\tan \theta = \frac{\frac{3}{5}}{\frac{4}{5}}$$

Simplifying the equation, we get:

$$\tan \theta = \frac{3}{4}$$

Substituting the Value of $\tan \theta$ into the Given Equation

Now that we have the value of $\tan \theta$, we can substitute it into the given equation:

$$\tan \theta = \frac{s^2}{49}$$

Substituting the value of $\tan \theta = \frac{3}{4}$, we get:

$$\frac{3}{4} = \frac{s^2}{49}$$

Solving for $s$

To solve for $s$, we can cross-multiply the equation:

$$3 \cdot 49 = 4 \cdot s^2$$

Simplifying the equation, we get:

$$147 = 4 \cdot s^2$$

Dividing both sides by 4, we get:

$$\frac{147}{4} = s^2$$

Taking the square root of both sides, we get:

$$s = \pm \sqrt{\frac{147}{4}}$$

Since the speed $s$ is positive, we take the positive square root:

$$s = \frac{\sqrt{147}}{2}$$

Conclusion

In this article, we guided a student through the process of solving a trigonometric equation involving speed and angle of incline. We used the Pythagorean identity to find the value of $\cos \theta$, and then used the definition of the tangent function to find the value of $\tan \theta$. We then substituted the value of $\tan \theta$ into the given equation and solved for $s$. The final answer is $\boxed{\frac{\sqrt{147}}{2}}$.

Final Answer

The final answer is $\boxed{\frac{\sqrt{147}}{2}}$.

Introduction

In our previous article, we guided a student through the process of solving a trigonometric equation involving speed and angle of incline. We used the Pythagorean identity to find the value of $\cos \theta$, and then used the definition of the tangent function to find the value of $\tan \theta$. We then substituted the value of $\tan \theta$ into the given equation and solved for $s$. In this article, we will provide a Q&A section to help students better understand the solution and address any questions they may have.

Q&A

Q: What is the Pythagorean identity, and how is it used in this problem?

A: The Pythagorean identity is a fundamental concept in trigonometry that states $\sin^2 \theta + \cos^2 \theta = 1$. We use this identity to find the value of $\cos \theta$ when we are given the value of $\sin \theta$. In this problem, we are given $\sin \theta = \frac{3}{5}$, so we can use the Pythagorean identity to find $\cos \theta = \frac{4}{5}$.

Q: Why do we take the positive square root of $\cos^2 \theta$?

A: We take the positive square root of $\cos^2 \theta$ because the angle of incline $\theta$ is positive. If we were dealing with a negative angle, we would take the negative square root.

Q: How do we find the value of $\tan \theta$?

A: We find the value of $\tan \theta$ using the definition of the tangent function: $\tan \theta = \frac{\sin \theta}{\cos \theta}$. In this problem, we are given $\sin \theta = \frac{3}{5}$ and $\cos \theta = \frac{4}{5}$, so we can substitute these values into the definition of the tangent function to find $\tan \theta = \frac{3}{4}$.

Q: Why do we cross-multiply the equation $\frac{3}{4} = \frac{s^2}{49}$?

A: We cross-multiply the equation $\frac{3}{4} = \frac{s^2}{49}$ to solve for $s$. By cross-multiplying, we can eliminate the fractions and solve for $s$.

Q: What is the final answer, and how do we interpret it?

A: The final answer is $\boxed{\frac{\sqrt{147}}{2}}$. This means that the speed $s$ of the toy car is $\frac{\sqrt{147}}{2}$ feet per second.

Common Mistakes to Avoid

1. Not using the Pythagorean identity to find the value of $\cos \theta$

Make sure to use the Pythagorean identity to find the value of $\cos \theta$ when you are given the value of $\sin \theta$.

2. Not taking the positive square root of $\cos^2 \theta$

Make sure to take the positive square root of $\cos^2 \theta$ when the angle of incline $\theta$ is positive.

3. Not using the definition of the tangent function to find the value of $\tan \theta$

Make sure to use the definition of the tangent function to find the value of $\tan \theta$ when you are given the values of $\sin \theta$ and $\cos \theta$.

Conclusion

In this Q&A article, we addressed common questions and misconceptions about solving a trigonometric equation involving speed and angle of incline. We hope that this article has helped students better understand the solution and address any questions they may have. Remember to use the Pythagorean identity to find the value of $\cos \theta$, take the positive square root of $\cos^2 \theta$, and use the definition of the tangent function to find the value of $\tan \theta$.