Use A Half-angle Formula To Find The Exact Value Of { \cos 67.5^\circ$}$.

Introduction

Trigonometric functions are essential in mathematics, and understanding how to find their exact values is crucial for solving various mathematical problems. One of the techniques used to find exact values of trigonometric functions is the half-angle formula. In this article, we will explore how to use the half-angle formula to find the exact value of $\cos 67.5^\circ$.

What is the Half-Angle Formula?

The half-angle formula is a technique used to find the exact value of a trigonometric function when the angle is halved. It is based on the identity $\sin^2 x + \cos^2 x = 1$. The half-angle formula for cosine is given by:

$$\cos \frac{x}{2} = \pm \sqrt{\frac{1 + \cos x}{2}}$$

Applying the Half-Angle Formula to Find $\cos 67.5^\circ$

To find the exact value of $\cos 67.5^\circ$, we can use the half-angle formula. First, we need to find the value of $\cos 135^\circ$. We know that $\cos 135^\circ = -\cos (180^\circ - 135^\circ) = -\cos 45^\circ$. Since $\cos 45^\circ = \frac{1}{\sqrt{2}}$, we have:

$$\cos 135^\circ = -\frac{1}{\sqrt{2}}$$

Now, we can use the half-angle formula to find the value of $\cos 67.5^\circ$:

$$\cos 67.5^\circ = \cos \frac{135^\circ}{2} = \pm \sqrt{\frac{1 + \cos 135^\circ}{2}}$$

Substituting the value of $\cos 135^\circ$, we get:

$$\cos 67.5^\circ = \pm \sqrt{\frac{1 - \frac{1}{\sqrt{2}}}{2}}$$

Simplifying the Expression

To simplify the expression, we can rationalize the denominator by multiplying the numerator and denominator by $\sqrt{2}$:

$$\cos 67.5^\circ = \pm \sqrt{\frac{\sqrt{2} - 1}{2\sqrt{2}}}$$

Simplifying further, we get:

$$\cos 67.5^\circ = \pm \sqrt{\frac{\sqrt{2} - 1}{2\sqrt{2}}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$$

$$\cos 67.5^\circ = \pm \frac{\sqrt{2}(\sqrt{2} - 1)}{2\sqrt{2}}$$

$$\cos 67.5^\circ = \pm \frac{2 - \sqrt{2}}{2\sqrt{2}}$$

Rationalizing the Denominator

To rationalize the denominator, we can multiply the numerator and denominator by $\sqrt{2}$:

$$\cos 67.5^\circ = \pm \frac{(2 - \sqrt{2})\sqrt{2}}{2\sqrt{2}\sqrt{2}}$$

Simplifying further, we get:

$$\cos 67.5^\circ = \pm \frac{2\sqrt{2} - 2}{4}$$

Simplifying the Expression

To simplify the expression, we can factor out a $2$ from the numerator:

$$\cos 67.5^\circ = \pm \frac{2(2\sqrt{2} - 1)}{4}$$

Simplifying further, we get:

$$\cos 67.5^\circ = \pm \frac{2\sqrt{2} - 1}{2}$$

Conclusion

In this article, we used the half-angle formula to find the exact value of $\cos 67.5^\circ$. We first found the value of $\cos 135^\circ$ and then used the half-angle formula to find the value of $\cos 67.5^\circ$. We simplified the expression by rationalizing the denominator and factoring out a $2$ from the numerator. The final answer is $\pm \frac{2\sqrt{2} - 1}{2}$.

Final Answer

The final answer is $\boxed{\pm \frac{2\sqrt{2} - 1}{2}}$.

Common Mistakes to Avoid

When using the half-angle formula, it is essential to remember that the formula is based on the identity $\sin^2 x + \cos^2 x = 1$. This means that the formula only works when the angle is halved. Additionally, when simplifying the expression, it is crucial to rationalize the denominator and factor out any common factors.

Real-World Applications

The half-angle formula has numerous real-world applications in fields such as physics, engineering, and computer science. For example, in physics, the half-angle formula is used to find the exact value of the cosine function when the angle is halved. In engineering, the half-angle formula is used to design and optimize systems that involve trigonometric functions. In computer science, the half-angle formula is used to develop algorithms that involve trigonometric functions.

Future Research Directions

Future research directions in the area of half-angle formulas include developing new techniques for finding exact values of trigonometric functions and exploring the applications of half-angle formulas in various fields. Additionally, researchers can investigate the use of half-angle formulas in machine learning and artificial intelligence.

Conclusion

In conclusion, the half-angle formula is a powerful technique for finding exact values of trigonometric functions. By using the half-angle formula, we can find the exact value of $\cos 67.5^\circ$. The final answer is $\boxed{\pm \frac{2\sqrt{2} - 1}{2}}$.

Q: What is the half-angle formula?

A: The half-angle formula is a technique used to find the exact value of a trigonometric function when the angle is halved. It is based on the identity $\sin^2 x + \cos^2 x = 1$.

Q: How do I apply the half-angle formula?

A: To apply the half-angle formula, you need to find the value of $\cos x$ and then use the formula $\cos \frac{x}{2} = \pm \sqrt{\frac{1 + \cos x}{2}}$.

Q: What are the common mistakes to avoid when using the half-angle formula?

A: When using the half-angle formula, it is essential to remember that the formula is based on the identity $\sin^2 x + \cos^2 x = 1$. This means that the formula only works when the angle is halved. Additionally, when simplifying the expression, it is crucial to rationalize the denominator and factor out any common factors.

Q: What are the real-world applications of the half-angle formula?

A: The half-angle formula has numerous real-world applications in fields such as physics, engineering, and computer science. For example, in physics, the half-angle formula is used to find the exact value of the cosine function when the angle is halved. In engineering, the half-angle formula is used to design and optimize systems that involve trigonometric functions. In computer science, the half-angle formula is used to develop algorithms that involve trigonometric functions.

Q: Can I use the half-angle formula to find the exact value of any trigonometric function?

A: No, the half-angle formula only works for cosine and sine functions. However, you can use the half-angle formula to find the exact value of other trigonometric functions by using the relationships between the trigonometric functions.

Q: How do I simplify the expression when using the half-angle formula?

A: To simplify the expression, you need to rationalize the denominator and factor out any common factors. You can also use algebraic manipulations to simplify the expression.

Q: What are the limitations of the half-angle formula?

A: The half-angle formula has several limitations. For example, it only works when the angle is halved, and it can be difficult to apply the formula when the angle is not a multiple of 30 or 45 degrees.

Q: Can I use the half-angle formula to find the exact value of a trigonometric function when the angle is not a multiple of 30 or 45 degrees?

A: No, the half-angle formula only works when the angle is a multiple of 30 or 45 degrees. However, you can use other techniques, such as the sum and difference formulas, to find the exact value of a trigonometric function when the angle is not a multiple of 30 or 45 degrees.

Q: How do I choose between the half-angle formula and other techniques for finding the exact value of a trigonometric function?

A: When choosing between the half-angle formula and other techniques, you need to consider the specific problem and the angle involved. If the angle is a multiple of 30 or 45 degrees, the half-angle formula may be the best choice. However, if the angle is not a multiple of 30 or 45 degrees, you may need to use other techniques, such as the sum and difference formulas.

Q: What are the benefits of using the half-angle formula?

A: The half-angle formula has several benefits. For example, it can be used to find the exact value of a trigonometric function when the angle is halved, and it can be used to simplify complex expressions involving trigonometric functions.

Q: What are the challenges of using the half-angle formula?

A: The half-angle formula can be challenging to apply, especially when the angle is not a multiple of 30 or 45 degrees. Additionally, the formula can be difficult to simplify, and it may require the use of algebraic manipulations.

Q: Can I use the half-angle formula to find the exact value of a trigonometric function in a right triangle?

A: Yes, the half-angle formula can be used to find the exact value of a trigonometric function in a right triangle. However, you need to use the formula carefully and consider the specific problem and the angle involved.

Q: What are the applications of the half-angle formula in calculus?

A: The half-angle formula has several applications in calculus. For example, it can be used to find the exact value of a trigonometric function when the angle is halved, and it can be used to simplify complex expressions involving trigonometric functions.

Q: Can I use the half-angle formula to find the exact value of a trigonometric function in a parametric equation?

A: Yes, the half-angle formula can be used to find the exact value of a trigonometric function in a parametric equation. However, you need to use the formula carefully and consider the specific problem and the angle involved.

Q: What are the limitations of the half-angle formula in parametric equations?

A: The half-angle formula has several limitations in parametric equations. For example, it only works when the angle is a multiple of 30 or 45 degrees, and it can be difficult to apply the formula when the angle is not a multiple of 30 or 45 degrees.

Q: Can I use the half-angle formula to find the exact value of a trigonometric function in a polar equation?

A: Yes, the half-angle formula can be used to find the exact value of a trigonometric function in a polar equation. However, you need to use the formula carefully and consider the specific problem and the angle involved.

Q: What are the applications of the half-angle formula in polar equations?

A: The half-angle formula has several applications in polar equations. For example, it can be used to find the exact value of a trigonometric function when the angle is halved, and it can be used to simplify complex expressions involving trigonometric functions.

Q: Can I use the half-angle formula to find the exact value of a trigonometric function in a parametric polar equation?

A: Yes, the half-angle formula can be used to find the exact value of a trigonometric function in a parametric polar equation. However, you need to use the formula carefully and consider the specific problem and the angle involved.

Q: What are the limitations of the half-angle formula in parametric polar equations?

A: The half-angle formula has several limitations in parametric polar equations. For example, it only works when the angle is a multiple of 30 or 45 degrees, and it can be difficult to apply the formula when the angle is not a multiple of 30 or 45 degrees.

Q: Can I use the half-angle formula to find the exact value of a trigonometric function in a three-dimensional equation?

A: Yes, the half-angle formula can be used to find the exact value of a trigonometric function in a three-dimensional equation. However, you need to use the formula carefully and consider the specific problem and the angle involved.

Q: What are the applications of the half-angle formula in three-dimensional equations?

A: The half-angle formula has several applications in three-dimensional equations. For example, it can be used to find the exact value of a trigonometric function when the angle is halved, and it can be used to simplify complex expressions involving trigonometric functions.

Q: Can I use the half-angle formula to find the exact value of a trigonometric function in a parametric three-dimensional equation?

A: Yes, the half-angle formula can be used to find the exact value of a trigonometric function in a parametric three-dimensional equation. However, you need to use the formula carefully and consider the specific problem and the angle involved.

Q: What are the limitations of the half-angle formula in parametric three-dimensional equations?

A: The half-angle formula has several limitations in parametric three-dimensional equations. For example, it only works when the angle is a multiple of 30 or 45 degrees, and it can be difficult to apply the formula when the angle is not a multiple of 30 or 45 degrees.

Conclusion

In conclusion, the half-angle formula is a powerful technique for finding exact values of trigonometric functions. By using the half-angle formula, you can find the exact value of a trigonometric function when the angle is halved. However, the formula has several limitations, and it can be difficult to apply in certain situations.