Use A Sum Or Difference Formula To Find The Exact Value Of The Trigonometric Function.$\[ \cos 165^{\circ} = \square \\](Simplify Your Answer, Including Any Radicals. Use Integers Or Fractions For Any Numbers In The Expression. Rationalize All

Introduction

Trigonometric functions are essential in mathematics, particularly in trigonometry and calculus. These functions describe the relationships between the sides and angles of triangles. In this article, we will focus on using sum or difference formulas to find the exact values of trigonometric functions. We will use the cosine function as an example and explore how to apply the sum or difference formulas to find the exact value of $\cos 165^{\circ}$.

Understanding Sum and Difference Formulas

The sum and difference formulas for trigonometric functions are used to express the values of trigonometric functions in terms of other angles. These formulas are essential in solving trigonometric equations and finding the exact values of trigonometric functions.

The sum formula for cosine is given by:

$$\cos (a + b) = \cos a \cos b - \sin a \sin b$$

The difference formula for cosine is given by:

$$\cos (a - b) = \cos a \cos b + \sin a \sin b$$

Applying Sum or Difference Formulas to Find Exact Values

To find the exact value of $\cos 165^{\circ}$, we can use the sum or difference formulas. We can rewrite $165^{\circ}$ as the sum of $180^{\circ}$ and $-15^{\circ}$.

Using the sum formula for cosine, we have:

$$\cos (180^{\circ} - 15^{\circ}) = \cos 180^{\circ} \cos (-15^{\circ}) - \sin 180^{\circ} \sin (-15^{\circ})$$

We know that $\cos 180^{\circ} = -1$ and $\sin 180^{\circ} = 0$. Therefore, we have:

$$\cos (180^{\circ} - 15^{\circ}) = (-1) \cos (-15^{\circ}) - (0) \sin (-15^{\circ})$$

$$\cos (180^{\circ} - 15^{\circ}) = -\cos (-15^{\circ})$$

Finding the Exact Value of $\cos (-15^{\circ})$

To find the exact value of $\cos (-15^{\circ})$, we can use the fact that $\cos (-\theta) = \cos \theta$. Therefore, we have:

$$\cos (-15^{\circ}) = \cos 15^{\circ}$$

Finding the Exact Value of $\cos 15^{\circ}$

To find the exact value of $\cos 15^{\circ}$, we can use the sum formula for cosine:

$$\cos (a + b) = \cos a \cos b - \sin a \sin b$$

We can rewrite $15^{\circ}$ as the sum of $45^{\circ}$ and $-30^{\circ}$.

Using the sum formula for cosine, we have:

$$\cos (45^{\circ} - 30^{\circ}) = \cos 45^{\circ} \cos (-30^{\circ}) - \sin 45^{\circ} \sin (-30^{\circ})$$

We know that $\cos 45^{\circ} = \frac{\sqrt{2}}{2}$, $\sin 45^{\circ} = \frac{\sqrt{2}}{2}$, and $\cos (-30^{\circ}) = \cos 30^{\circ} = \frac{\sqrt{3}}{2}$. Therefore, we have:

$$\cos (45^{\circ} - 30^{\circ}) = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right) \left(-\frac{1}{2}\right)$$

$$\cos (45^{\circ} - 30^{\circ}) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}$$

Finding the Exact Value of $\cos 165^{\circ}$

Now that we have found the exact value of $\cos 15^{\circ}$, we can find the exact value of $\cos 165^{\circ}$.

Using the fact that $\cos (180^{\circ} - 15^{\circ}) = -\cos (-15^{\circ})$, we have:

$$\cos 165^{\circ} = -\cos (-15^{\circ})$$

$$\cos 165^{\circ} = -\cos 15^{\circ}$$

$$\cos 165^{\circ} = -\left(\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}\right)$$

$$\cos 165^{\circ} = -\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$$

Conclusion

In this article, we used the sum or difference formulas to find the exact value of $\cos 165^{\circ}$. We rewrote $165^{\circ}$ as the sum of $180^{\circ}$ and $-15^{\circ}$ and used the sum formula for cosine to find the exact value of $\cos 165^{\circ}$. We also used the fact that $\cos (-\theta) = \cos \theta$ to find the exact value of $\cos (-15^{\circ})$. Finally, we used the sum formula for cosine to find the exact value of $\cos 15^{\circ}$.

The exact value of $\cos 165^{\circ}$ is $-\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$. This value can be used to solve trigonometric equations and find the exact values of trigonometric functions.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Trigonometric Functions" by Wolfram MathWorld

Glossary

• Sum formula: A formula used to express the value of a trigonometric function in terms of other angles.
• Difference formula: A formula used to express the value of a trigonometric function in terms of other angles.
• Cosine function: A trigonometric function that describes the ratio of the adjacent side to the hypotenuse in a right triangle.
• Trigonometric equation: An equation that involves trigonometric functions.
• Exact value: A value that is exact and not an approximation.
  

Q: What is the sum formula for cosine?

A: The sum formula for cosine is given by:

$$\cos (a + b) = \cos a \cos b - \sin a \sin b$$

Q: What is the difference formula for cosine?

A: The difference formula for cosine is given by:

$$\cos (a - b) = \cos a \cos b + \sin a \sin b$$

Q: How do I use the sum or difference formulas to find the exact value of a trigonometric function?

A: To use the sum or difference formulas to find the exact value of a trigonometric function, you need to:

1. Rewrite the angle in terms of other angles using the sum or difference formulas.
2. Substitute the values of the trigonometric functions for the new angles into the formula.
3. Simplify the expression to find the exact value of the trigonometric function.

Q: What is the exact value of $\cos 165^{\circ}$?

A: The exact value of $\cos 165^{\circ}$ is $-\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$.

Q: How do I find the exact value of $\cos (-15^{\circ})$?

A: To find the exact value of $\cos (-15^{\circ})$, you can use the fact that $\cos (-\theta) = \cos \theta$. Therefore, $\cos (-15^{\circ}) = \cos 15^{\circ}$.

Q: How do I find the exact value of $\cos 15^{\circ}$?

A: To find the exact value of $\cos 15^{\circ}$, you can use the sum formula for cosine:

$$\cos (a + b) = \cos a \cos b - \sin a \sin b$$

Rewrite $15^{\circ}$ as the sum of $45^{\circ}$ and $-30^{\circ}$ and substitute the values of the trigonometric functions into the formula.

Q: What are some common mistakes to avoid when using the sum or difference formulas?

A: Some common mistakes to avoid when using the sum or difference formulas include:

• Not rewriting the angle in terms of other angles using the sum or difference formulas.
• Not substituting the values of the trigonometric functions for the new angles into the formula.
• Not simplifying the expression to find the exact value of the trigonometric function.

Q: How do I know which formula to use, the sum formula or the difference formula?

A: To determine which formula to use, you need to rewrite the angle in terms of other angles using the sum or difference formulas. If the angle is the sum of two angles, use the sum formula. If the angle is the difference of two angles, use the difference formula.

Q: Can I use the sum or difference formulas to find the exact value of any trigonometric function?

A: Yes, you can use the sum or difference formulas to find the exact value of any trigonometric function. However, you need to rewrite the angle in terms of other angles using the sum or difference formulas and substitute the values of the trigonometric functions into the formula.

Q: What are some real-world applications of using the sum or difference formulas to find exact values of trigonometric functions?

A: Some real-world applications of using the sum or difference formulas to find exact values of trigonometric functions include:

• Solving trigonometric equations in physics and engineering.
• Finding the exact values of trigonometric functions in calculus and differential equations.
• Using trigonometric functions to model real-world phenomena, such as the motion of objects in physics and engineering.

Q: How do I practice using the sum or difference formulas to find exact values of trigonometric functions?

A: To practice using the sum or difference formulas to find exact values of trigonometric functions, you can:

• Work on problems that involve finding the exact value of a trigonometric function using the sum or difference formulas.
• Use online resources, such as calculators and worksheets, to practice using the sum or difference formulas.
• Ask your teacher or tutor for help and guidance on using the sum or difference formulas to find exact values of trigonometric functions.