Given That Sin ⁡ ( 5 Π 18 ) ≈ 0.7660 \sin \left(\frac{5 \pi}{18}\right) \approx 0.7660 Sin ( 18 5 Π ​ ) ≈ 0.7660 And Cos ⁡ ( 5 Π 18 ) ≈ 0.6428 \cos \left(\frac{5 \pi}{18}\right) \approx 0.6428 Cos ( 18 5 Π ​ ) ≈ 0.6428 , Use These Approximate Function Values To Find:a) The Other Four Trigonometric Function Values For $\frac{5

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will explore the concept of trigonometric function values and how to find them using given approximate values.

Given Approximate Function Values

We are given the following approximate function values:

• $\sin \left(\frac{5 \pi}{18}\right) \approx 0.7660$
• $\cos \left(\frac{5 \pi}{18}\right) \approx 0.6428$

These values are approximate, meaning they are not exact but rather close to the actual values.

The Other Four Trigonometric Function Values

Using the given approximate function values, we can find the other four trigonometric function values for $\frac{5 \pi}{18}$.

a) Tangent Function Value

The tangent function value is defined as the ratio of the sine and cosine function values.

$\tan \left(\frac{5 \pi}{18}\right) = \frac{\sin \left(\frac{5 \pi}{18}\right)}{\cos \left(\frac{5 \pi}{18}\right)}$

Substituting the given approximate function values, we get:

$\tan \left(\frac{5 \pi}{18}\right) = \frac{0.7660}{0.6428} \approx 1.1943$

b) Cotangent Function Value

The cotangent function value is the reciprocal of the tangent function value.

$\cot \left(\frac{5 \pi}{18}\right) = \frac{1}{\tan \left(\frac{5 \pi}{18}\right)}$

Substituting the approximate tangent function value, we get:

$\cot \left(\frac{5 \pi}{18}\right) = \frac{1}{1.1943} \approx 0.8353$

c) Secant Function Value

The secant function value is the reciprocal of the cosine function value.

$\sec \left(\frac{5 \pi}{18}\right) = \frac{1}{\cos \left(\frac{5 \pi}{18}\right)}$

Substituting the given approximate cosine function value, we get:

$\sec \left(\frac{5 \pi}{18}\right) = \frac{1}{0.6428} \approx 1.5545$

d) Cosecant Function Value

The cosecant function value is the reciprocal of the sine function value.

$\csc \left(\frac{5 \pi}{18}\right) = \frac{1}{\sin \left(\frac{5 \pi}{18}\right)}$

Substituting the given approximate sine function value, we get:

$\csc \left(\frac{5 \pi}{18}\right) = \frac{1}{0.7660} \approx 1.3043$

Conclusion

In this article, we have explored the concept of trigonometric function values and how to find them using given approximate values. We have used the given approximate function values to find the other four trigonometric function values for $\frac{5 \pi}{18}$. The approximate function values are:

• $\sin \left(\frac{5 \pi}{18}\right) \approx 0.7660$
• $\cos \left(\frac{5 \pi}{18}\right) \approx 0.6428$
• $\tan \left(\frac{5 \pi}{18}\right) \approx 1.1943$
• $\cot \left(\frac{5 \pi}{18}\right) \approx 0.8353$
• $\sec \left(\frac{5 \pi}{18}\right) \approx 1.5545$
• $\csc \left(\frac{5 \pi}{18}\right) \approx 1.3043$

These values are approximate and can be used in various applications in mathematics and other fields.

References

• [1] "Trigonometry" by Michael Corral, 2019.
• [2] "Trigonometric Functions" by Math Open Reference, 2020.

Note

Q&A: Trigonometric Function Values

Frequently Asked Questions

Q1: What are trigonometric function values?

A1: Trigonometric function values are the values of the six basic trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant) for a given angle.

Q2: How are trigonometric function values used in real-life applications?

A2: Trigonometric function values are used in various real-life applications, including navigation, physics, engineering, and architecture. They are used to calculate distances, heights, and angles in different situations.

Q3: What is the difference between sine and cosine function values?

A3: The sine function value is the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle, while the cosine function value is the ratio of the length of the adjacent side to the length of the hypotenuse.

Q4: How can I find the other four trigonometric function values using given approximate function values?

A4: To find the other four trigonometric function values, you can use the following formulas:

• $\tan \left(\theta\right) = \frac{\sin \left(\theta\right)}{\cos \left(\theta\right)}$
• $\cot \left(\theta\right) = \frac{1}{\tan \left(\theta\right)}$
• $\sec \left(\theta\right) = \frac{1}{\cos \left(\theta\right)}$
• $\csc \left(\theta\right) = \frac{1}{\sin \left(\theta\right)}$

Q5: What are the approximate function values for $\frac{5 \pi}{18}$?

A5: The approximate function values for $\frac{5 \pi}{18}$ are:

• $\sin \left(\frac{5 \pi}{18}\right) \approx 0.7660$
• $\cos \left(\frac{5 \pi}{18}\right) \approx 0.6428$
• $\tan \left(\frac{5 \pi}{18}\right) \approx 1.1943$
• $\cot \left(\frac{5 \pi}{18}\right) \approx 0.8353$
• $\sec \left(\frac{5 \pi}{18}\right) \approx 1.5545$
• $\csc \left(\frac{5 \pi}{18}\right) \approx 1.3043$

Q6: How can I use trigonometric function values in real-life applications?

A6: Trigonometric function values can be used in various real-life applications, including:

• Navigation: to calculate distances and angles between two points
• Physics: to calculate velocities and accelerations
• Engineering: to design and build structures
• Architecture: to design and build buildings

Conclusion

In this article, we have answered some frequently asked questions about trigonometric function values. We have discussed the concept of trigonometric function values, their applications, and how to find the other four trigonometric function values using given approximate function values. We have also provided some examples of real-life applications of trigonometric function values.

References

• [1] "Trigonometry" by Michael Corral, 2019.
• [2] "Trigonometric Functions" by Math Open Reference, 2020.

Note

The values obtained in this article are approximate and may vary slightly depending on the method used to calculate them.