If Sin ⁡ Θ ≈ − 0.7660 \sin \theta \approx -0.7660 Sin Θ ≈ − 0.7660 , Which Of The Following Represents An Approximate Value Of Tan ⁡ Θ \tan \theta Tan Θ For 180 ∘ \textless Θ \textless 270 ∘ 180^{\circ} \ \textless \ \theta \ \textless \ 270^{\circ} 18 0 ∘ \textless Θ \textless 27 0 ∘ ?A. 0.7660 B. 0.8392 C. 1.1916 D. 1.4198

Introduction

Trigonometric functions are essential in mathematics, particularly in calculus and analytical geometry. These functions describe the relationships between the sides and angles of triangles. In this article, we will focus on approximating the value of the tangent function given the sine value of an angle. We will use the given information to determine the approximate value of the tangent function for a specific range of angles.

Understanding the Problem

The problem provides the approximate value of the sine function for an angle $\theta$, which is $\sin \theta \approx -0.7660$. We are asked to find the approximate value of the tangent function for the same angle, given that the angle lies in the range $180^{\circ} \ \textless \ \theta \ \textless \ 270^{\circ}$.

Recalling Trigonometric Identities

To solve this problem, we need to recall the trigonometric identity that relates the sine and tangent functions:

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

We can use this identity to express the tangent function in terms of the sine and cosine functions.

Finding the Cosine Value

Since we are given the sine value, we need to find the cosine value to use in the tangent identity. We can use the Pythagorean identity to find the cosine value:

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

Rearranging this equation to solve for the cosine value, we get:

$$\cos \theta = \pm \sqrt{1 - \sin^2 \theta}$$

Since the angle lies in the range $180^{\circ} \ \textless \ \theta \ \textless \ 270^{\circ}$, the cosine value will be negative.

Calculating the Cosine Value

Using the given sine value, we can calculate the cosine value:

$$\cos \theta = -\sqrt{1 - (-0.7660)^2}$$

$$\cos \theta = -\sqrt{1 - 0.5854}$$

$$\cos \theta = -\sqrt{0.4146}$$

$$\cos \theta \approx -0.6436$$

Finding the Tangent Value

Now that we have the sine and cosine values, we can use the tangent identity to find the tangent value:

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

$$\tan \theta = \frac{-0.7660}{-0.6436}$$

$$\tan \theta \approx 1.1916$$

Conclusion

In this article, we used the given sine value to find the approximate value of the tangent function for an angle in the range $180^{\circ} \ \textless \ \theta \ \textless \ 270^{\circ}$. We recalled the trigonometric identity that relates the sine and tangent functions, found the cosine value using the Pythagorean identity, and calculated the tangent value using the tangent identity. The approximate value of the tangent function is $\boxed{1.1916}$.

Answer

Introduction

In our previous article, we discussed how to approximate the value of the tangent function given the sine value of an angle. We used the given information to determine the approximate value of the tangent function for a specific range of angles. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topic.

Q: What is the relationship between the sine and tangent functions?

A: The tangent function is related to the sine and cosine functions through the following identity:

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

This identity allows us to express the tangent function in terms of the sine and cosine functions.

Q: How do I find the cosine value when given the sine value?

A: To find the cosine value when given the sine value, you can use the Pythagorean identity:

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

Rearranging this equation to solve for the cosine value, you get:

$$\cos \theta = \pm \sqrt{1 - \sin^2 \theta}$$

Q: What is the range of the cosine function?

A: The range of the cosine function is $-1 \leq \cos \theta \leq 1$. This means that the cosine value can be either positive or negative, depending on the angle.

Q: How do I determine the sign of the cosine value?

A: To determine the sign of the cosine value, you need to consider the quadrant in which the angle lies. If the angle lies in the first or fourth quadrant, the cosine value is positive. If the angle lies in the second or third quadrant, the cosine value is negative.

Q: Can I use the tangent identity to find the sine value?

A: Yes, you can use the tangent identity to find the sine value. Rearranging the tangent identity to solve for the sine value, you get:

$$\sin \theta = \tan \theta \cdot \cos \theta$$

Q: What is the relationship between the tangent and cotangent functions?

A: The cotangent function is the reciprocal of the tangent function:

$$\cot \theta = \frac{1}{\tan \theta}$$

This means that the cotangent function is equal to the reciprocal of the tangent function.

Q: Can I use the tangent identity to find the cotangent value?

A: Yes, you can use the tangent identity to find the cotangent value. Rearranging the tangent identity to solve for the cotangent value, you get:

$$\cot \theta = \frac{1}{\tan \theta}$$

Conclusion

In this Q&A article, we provided answers to common questions related to the tangent function and its relationship to the sine and cosine functions. We also discussed how to find the cosine value when given the sine value, and how to determine the sign of the cosine value. We hope that this article has helped to clarify any doubts and provide additional information on the topic.

Additional Resources

For more information on trigonometric functions, we recommend the following resources:

• Khan Academy: Trigonometry
• MIT OpenCourseWare: Trigonometry
• Wolfram MathWorld: Trigonometry

Answer Key

1. The relationship between the sine and tangent functions is given by the identity $\tan \theta = \frac{\sin \theta}{\cos \theta}$.
2. To find the cosine value when given the sine value, use the Pythagorean identity: $\cos \theta = \pm \sqrt{1 - \sin^2 \theta}$.
3. The range of the cosine function is $-1 \leq \cos \theta \leq 1$.
4. To determine the sign of the cosine value, consider the quadrant in which the angle lies.
5. Yes, you can use the tangent identity to find the sine value: $\sin \theta = \tan \theta \cdot \cos \theta$.
6. The cotangent function is the reciprocal of the tangent function: $\cot \theta = \frac{1}{\tan \theta}$.
7. Yes, you can use the tangent identity to find the cotangent value: $\cot \theta = \frac{1}{\tan \theta}$.