4. If $\cos 36^{\circ} = K$, Determine In Terms Of $k$:4.1 $\cos \left(-36^{\circ}\right$\]4.2 $\sin 18^{\circ}$

Introduction

In this article, we will explore the relationship between the cosine of 36 degrees and the sine of 18 degrees. We will use the given value of $\cos 36^{\circ} = k$ to determine the values of $\cos \left(-36^{\circ}\right)$ and $\sin 18^{\circ}$ in terms of $k$.

4.1 Determine the Value of $\cos \left(-36^{\circ}\right)$

The cosine function is an even function, which means that $\cos (-x) = \cos x$. Therefore, we can write:

$$\cos \left(-36^{\circ}\right) = \cos 36^{\circ} = k$$

This means that the value of $\cos \left(-36^{\circ}\right)$ is equal to the value of $\cos 36^{\circ}$, which is given as $k$.

4.2 Determine the Value of $\sin 18^{\circ}$

To determine the value of $\sin 18^{\circ}$, we can use the fact that $\sin x = \cos (90^{\circ} - x)$. Therefore, we can write:

$$\sin 18^{\circ} = \cos (90^{\circ} - 18^{\circ}) = \cos 72^{\circ}$$

Now, we need to find the value of $\cos 72^{\circ}$ in terms of $k$. To do this, we can use the fact that $\cos (2x) = 2\cos^2 x - 1$. Let's set $x = 36^{\circ}$ and use the given value of $\cos 36^{\circ} = k$:

$$\cos 72^{\circ} = \cos (2 \cdot 36^{\circ}) = 2\cos^2 36^{\circ} - 1 = 2k^2 - 1$$

This means that the value of $\sin 18^{\circ}$ is equal to $2k^2 - 1$.

Conclusion

In this article, we have determined the values of $\cos \left(-36^{\circ}\right)$ and $\sin 18^{\circ}$ in terms of $k$. We have shown that $\cos \left(-36^{\circ}\right) = k$ and $\sin 18^{\circ} = 2k^2 - 1$. These results demonstrate the relationship between the cosine of 36 degrees and the sine of 18 degrees.

**Relationship between $\cos 36^{\circ}$ and $\sin 18^{\circ}$

The values of $\cos 36^{\circ}$ and $\sin 18^{\circ}$ are related through the double angle formula for cosine. Let's set $x = 36^{\circ}$ and use the given value of $\cos 36^{\circ} = k$:

$$\cos 72^{\circ} = \cos (2 \cdot 36^{\circ}) = 2\cos^2 36^{\circ} - 1 = 2k^2 - 1$$

This means that the value of $\sin 18^{\circ}$ is equal to $2k^2 - 1$.

Properties of the Golden Ratio

The golden ratio, denoted by $\phi$, is an irrational number that is approximately equal to 1.61803398875. The golden ratio has many interesting properties, including the fact that it is the solution to the equation $\phi^2 = \phi + 1$. The golden ratio is also related to the Fibonacci sequence, which is a sequence of numbers in which each number is the sum of the two preceding numbers.

The golden ratio is also related to the values of $\cos 36^{\circ}$ and $\sin 18^{\circ}$. Let's set $x = 36^{\circ}$ and use the given value of $\cos 36^{\circ} = k$:

$$\cos 72^{\circ} = \cos (2 \cdot 36^{\circ}) = 2\cos^2 36^{\circ} - 1 = 2k^2 - 1$$

This means that the value of $\sin 18^{\circ}$ is equal to $2k^2 - 1$.

Conclusion

In this article, we have determined the values of $\cos \left(-36^{\circ}\right)$ and $\sin 18^{\circ}$ in terms of $k$. We have shown that $\cos \left(-36^{\circ}\right) = k$ and $\sin 18^{\circ} = 2k^2 - 1$. These results demonstrate the relationship between the cosine of 36 degrees and the sine of 18 degrees.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Geometry" by I.M. Gelfand

Appendix

The following is a list of formulas and identities that are used in this article:

• $\cos (2x) = 2\cos^2 x - 1$
• $\sin x = \cos (90^{\circ} - x)$
• $\cos (-x) = \cos x$

$$$$$$

Introduction

In our previous article, we determined the values of $\cos \left(-36^{\circ}\right)$ and $\sin 18^{\circ}$ in terms of $k$. We showed that $\cos \left(-36^{\circ}\right) = k$ and $\sin 18^{\circ} = 2k^2 - 1$. In this article, we will answer some common questions related to these results.

Q: What is the relationship between $\cos 36^{\circ}$ and $\sin 18^{\circ}$?

A: The values of $\cos 36^{\circ}$ and $\sin 18^{\circ}$ are related through the double angle formula for cosine. Let's set $x = 36^{\circ}$ and use the given value of $\cos 36^{\circ} = k$:

$$\cos 72^{\circ} = \cos (2 \cdot 36^{\circ}) = 2\cos^2 36^{\circ} - 1 = 2k^2 - 1$$

This means that the value of $\sin 18^{\circ}$ is equal to $2k^2 - 1$.

Q: How do you determine the value of $\cos \left(-36^{\circ}\right)$?

A: The cosine function is an even function, which means that $\cos (-x) = \cos x$. Therefore, we can write:

$$\cos \left(-36^{\circ}\right) = \cos 36^{\circ} = k$$

This means that the value of $\cos \left(-36^{\circ}\right)$ is equal to the value of $\cos 36^{\circ}$, which is given as $k$.

Q: What is the significance of the golden ratio in this context?

A: The golden ratio, denoted by $\phi$, is an irrational number that is approximately equal to 1.61803398875. The golden ratio has many interesting properties, including the fact that it is the solution to the equation $\phi^2 = \phi + 1$. The golden ratio is also related to the Fibonacci sequence, which is a sequence of numbers in which each number is the sum of the two preceding numbers.

The golden ratio is also related to the values of $\cos 36^{\circ}$ and $\sin 18^{\circ}$. Let's set $x = 36^{\circ}$ and use the given value of $\cos 36^{\circ} = k$:

$$\cos 72^{\circ} = \cos (2 \cdot 36^{\circ}) = 2\cos^2 36^{\circ} - 1 = 2k^2 - 1$$

This means that the value of $\sin 18^{\circ}$ is equal to $2k^2 - 1$.

Q: How do you use the double angle formula for cosine to determine the value of $\sin 18^{\circ}$?

A: To determine the value of $\sin 18^{\circ}$, we can use the fact that $\sin x = \cos (90^{\circ} - x)$. Therefore, we can write:

$$\sin 18^{\circ} = \cos (90^{\circ} - 18^{\circ}) = \cos 72^{\circ}$$

Now, we can use the double angle formula for cosine to determine the value of $\cos 72^{\circ}$:

$$\cos 72^{\circ} = \cos (2 \cdot 36^{\circ}) = 2\cos^2 36^{\circ} - 1 = 2k^2 - 1$$

This means that the value of $\sin 18^{\circ}$ is equal to $2k^2 - 1$.

Q: What are some common applications of trigonometry in real-world problems?

A: Trigonometry has many applications in real-world problems, including:

• Navigation: Trigonometry is used in navigation to determine the position and direction of objects.
• Physics: Trigonometry is used in physics to describe the motion of objects and the forces acting on them.
• Engineering: Trigonometry is used in engineering to design and build structures such as bridges and buildings.
• Computer Science: Trigonometry is used in computer science to create 3D graphics and animations.

Conclusion

In this article, we have answered some common questions related to the values of $\cos \left(-36^{\circ}\right)$ and $\sin 18^{\circ}$ in terms of $k$. We have shown that $\cos \left(-36^{\circ}\right) = k$ and $\sin 18^{\circ} = 2k^2 - 1$. These results demonstrate the relationship between the cosine of 36 degrees and the sine of 18 degrees.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Geometry" by I.M. Gelfand

Appendix

The following is a list of formulas and identities that are used in this article:

• $\cos (2x) = 2\cos^2 x - 1$
• $\sin x = \cos (90^{\circ} - x)$
• $\cos (-x) = \cos x$

These formulas and identities are used to derive the values of $\cos \left(-36^{\circ}\right)$ and $\sin 18^{\circ}$ in terms of $k$.