The Terminal Side Of An Angle In Standard Position Passes Through P ( − 3 , − 4 P(-3, -4 P ( − 3 , − 4 ]. What Is The Value Of Tan ⁡ Θ \tan \theta Tan Θ ?A. Tan ⁡ Θ = − 4 3 \tan \theta = -\frac{4}{3} Tan Θ = − 3 4 ​ B. Tan ⁡ Θ = − 3 4 \tan \theta = -\frac{3}{4} Tan Θ = − 4 3 ​ C. $\tan \theta =

Introduction

In trigonometry, the terminal side of an angle in standard position is a fundamental concept that helps us understand the relationships between the angle, the coordinates of a point on the terminal side, and the trigonometric functions. In this article, we will explore how to find the value of $\tan \theta$ when the terminal side of an angle in standard position passes through a given point.

Understanding the Terminal Side of an Angle

The terminal side of an angle in standard position is the side of the angle that lies in the coordinate plane and contains the vertex of the angle. The terminal side is defined by the coordinates of a point on the side, which can be represented as $(x, y)$. In this case, we are given that the terminal side of the angle passes through the point $P(-3, -4)$.

Recalling the Definition of $\tan \theta$

The tangent of an angle $\theta$ is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Mathematically, this can be represented as:

$$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$$

Finding the Value of $\tan \theta$

To find the value of $\tan \theta$, we need to determine the lengths of the sides opposite and adjacent to the angle. Since the terminal side of the angle passes through the point $P(-3, -4)$, we can use the coordinates of this point to find the lengths of the sides.

The length of the side opposite the angle is the distance from the origin $(0, 0)$ to the point $P(-3, -4)$. This can be calculated using the distance formula:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Plugging in the coordinates of the origin and the point $P$, we get:

$$d = \sqrt{(-3 - 0)^2 + (-4 - 0)^2}$$

$$d = \sqrt{9 + 16}$$

$$d = \sqrt{25}$$

$$d = 5$$

The length of the side adjacent to the angle is the distance from the origin $(0, 0)$ to the point where the terminal side intersects the x-axis. Since the terminal side passes through the point $P(-3, -4)$, the x-coordinate of this point is $-3$. Therefore, the length of the side adjacent to the angle is $|-3| = 3$.

Now that we have the lengths of the sides opposite and adjacent to the angle, we can find the value of $\tan \theta$:

$$\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{5}{3}$$

However, we are given that the terminal side of the angle passes through the point $P(-3, -4)$, which means that the angle is in the third quadrant. In the third quadrant, the tangent function is negative. Therefore, the value of $\tan \theta$ is:

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

Conclusion

In this article, we explored how to find the value of $\tan \theta$ when the terminal side of an angle in standard position passes through a given point. We used the coordinates of the point to find the lengths of the sides opposite and adjacent to the angle, and then used these lengths to calculate the value of $\tan \theta$. We found that the value of $\tan \theta$ is $-\frac{5}{3}$.

Discussion

The value of $\tan \theta$ is an important concept in trigonometry, and it has many real-world applications. For example, in engineering, the tangent function is used to calculate the slope of a line or the angle of a triangle. In physics, the tangent function is used to describe the motion of an object in a circular path.

Practice Problems

1. Find the value of $\tan \theta$ when the terminal side of an angle in standard position passes through the point $P(2, 3)$.
2. Find the value of $\tan \theta$ when the terminal side of an angle in standard position passes through the point $P(-2, 1)$.
3. Find the value of $\tan \theta$ when the terminal side of an angle in standard position passes through the point $P(1, -2)$.

Answer Key

1. $\tan \theta = \frac{3}{2}$
2. $\tan \theta = -\frac{1}{2}$
3. $\tan \theta = -\frac{2}{1}$
   Q&A: The Terminal Side of an Angle in Standard Position ===========================================================

Frequently Asked Questions

Q: What is the terminal side of an angle in standard position? A: The terminal side of an angle in standard position is the side of the angle that lies in the coordinate plane and contains the vertex of the angle.

Q: How do I find the value of $\tan \theta$ when the terminal side of an angle in standard position passes through a given point? A: To find the value of $\tan \theta$, you need to determine the lengths of the sides opposite and adjacent to the angle. You can use the coordinates of the point to find these lengths, and then use the tangent formula to calculate the value of $\tan \theta$.

Q: What is the tangent formula? A: The tangent formula is:

$$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$$

Q: How do I determine the lengths of the sides opposite and adjacent to the angle? A: You can use the distance formula to find the length of the side opposite the angle, and the x-coordinate of the point to find the length of the side adjacent to the angle.

Q: What is the distance formula? A: The distance formula is:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Q: How do I know which quadrant the angle is in? A: You can use the coordinates of the point to determine which quadrant the angle is in. If the x-coordinate is positive and the y-coordinate is positive, the angle is in the first quadrant. If the x-coordinate is negative and the y-coordinate is positive, the angle is in the second quadrant. If the x-coordinate is negative and the y-coordinate is negative, the angle is in the third quadrant. If the x-coordinate is positive and the y-coordinate is negative, the angle is in the fourth quadrant.

Q: How does the quadrant affect the value of $\tan \theta$? A: In the first and fourth quadrants, the value of $\tan \theta$ is positive. In the second and third quadrants, the value of $\tan \theta$ is negative.

Q: Can I use the tangent formula to find the value of $\tan \theta$ when the angle is in the second or third quadrant? A: Yes, you can use the tangent formula to find the value of $\tan \theta$ when the angle is in the second or third quadrant. However, you need to take into account the fact that the value of $\tan \theta$ is negative in these quadrants.

Q: How do I find the value of $\tan \theta$ when the angle is in the second or third quadrant? A: To find the value of $\tan \theta$ when the angle is in the second or third quadrant, you need to use the fact that the value of $\tan \theta$ is negative in these quadrants. You can do this by multiplying the value of $\tan \theta$ by $-1$.

Q: Can I use the tangent formula to find the value of $\tan \theta$ when the angle is in the fourth quadrant? A: Yes, you can use the tangent formula to find the value of $\tan \theta$ when the angle is in the fourth quadrant. However, you need to take into account the fact that the value of $\tan \theta$ is positive in this quadrant.

Q: How do I find the value of $\tan \theta$ when the angle is in the fourth quadrant? A: To find the value of $\tan \theta$ when the angle is in the fourth quadrant, you can use the tangent formula as usual.

Conclusion

In this article, we have answered some of the most frequently asked questions about the terminal side of an angle in standard position and the tangent function. We have covered topics such as how to find the value of $\tan \theta$ when the terminal side of an angle in standard position passes through a given point, how to determine the lengths of the sides opposite and adjacent to the angle, and how to use the tangent formula to find the value of $\tan \theta$ in different quadrants.

Practice Problems

1. Find the value of $\tan \theta$ when the terminal side of an angle in standard position passes through the point $P(2, 3)$.
2. Find the value of $\tan \theta$ when the terminal side of an angle in standard position passes through the point $P(-2, 1)$.
3. Find the value of $\tan \theta$ when the terminal side of an angle in standard position passes through the point $P(1, -2)$.

Answer Key

1. $\tan \theta = \frac{3}{2}$
2. $\tan \theta = -\frac{1}{2}$
3. $\tan \theta = -\frac{2}{1}$