Select The Correct Exact Answer:$\cos 80^{\circ} \cos 70^{\circ} - \sin 80^{\circ} \sin 70^{\circ}$A. $-\frac{1}{2}$ B. $\frac{1}{2}$ C. $\frac{\sqrt{3}}{2}$ D. $-\frac{\sqrt{3}}{2}$

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 focus on solving a specific trigonometric expression involving cosine and sine functions.

The Expression

The given expression is:

$$\cos 80^{\circ} \cos 70^{\circ} - \sin 80^{\circ} \sin 70^{\circ}$$

This expression involves the product of two cosine functions and the product of two sine functions. Our goal is to simplify this expression and find its exact value.

Using Trigonometric Identities

To simplify the given expression, we can use the trigonometric identity for the cosine of the difference of two angles:

$$\cos (A - B) = \cos A \cos B + \sin A \sin B$$

We can rewrite the given expression as:

$$\cos 80^{\circ} \cos 70^{\circ} - \sin 80^{\circ} \sin 70^{\circ} = -(\cos 80^{\circ} \cos 70^{\circ} + \sin 80^{\circ} \sin 70^{\circ})$$

Now, we can apply the trigonometric identity for the cosine of the difference of two angles:

$$\cos (A - B) = \cos A \cos B + \sin A \sin B$$

Substituting $A = 80^{\circ}$ and $B = 70^{\circ}$, we get:

$$\cos (80^{\circ} - 70^{\circ}) = \cos 80^{\circ} \cos 70^{\circ} + \sin 80^{\circ} \sin 70^{\circ}$$

Therefore, we can rewrite the given expression as:

$$\cos 80^{\circ} \cos 70^{\circ} - \sin 80^{\circ} \sin 70^{\circ} = -\cos (80^{\circ} - 70^{\circ})$$

Evaluating the Expression

Now, we can evaluate the expression by finding the value of $\cos (80^{\circ} - 70^{\circ})$.

$$\cos (80^{\circ} - 70^{\circ}) = \cos 10^{\circ}$$

Using a calculator or a trigonometric table, we can find that:

$$\cos 10^{\circ} = \frac{\sqrt{3}}{2}$$

Therefore, we can rewrite the given expression as:

$$\cos 80^{\circ} \cos 70^{\circ} - \sin 80^{\circ} \sin 70^{\circ} = -\frac{\sqrt{3}}{2}$$

Conclusion

In this article, we have solved a specific trigonometric expression involving cosine and sine functions. We used the trigonometric identity for the cosine of the difference of two angles to simplify the expression and find its exact value. The final answer is:

$$\cos 80^{\circ} \cos 70^{\circ} - \sin 80^{\circ} \sin 70^{\circ} = -\frac{\sqrt{3}}{2}$$

This result can be used in various applications, including physics, engineering, and navigation.

Answer

The correct answer is:

$$

Q: What is the trigonometric identity for the cosine of the difference of two angles?

A: The trigonometric identity for the cosine of the difference of two angles is:

$$\cos (A - B) = \cos A \cos B + \sin A \sin B$$

Q: How can I simplify the expression $\cos 80^{\circ} \cos 70^{\circ} - \sin 80^{\circ} \sin 70^{\circ}$?

A: To simplify the expression, you can use the trigonometric identity for the cosine of the difference of two angles. Rewrite the expression as:

$$\cos 80^{\circ} \cos 70^{\circ} - \sin 80^{\circ} \sin 70^{\circ} = -(\cos 80^{\circ} \cos 70^{\circ} + \sin 80^{\circ} \sin 70^{\circ})$$

Then, apply the trigonometric identity for the cosine of the difference of two angles:

$$\cos (A - B) = \cos A \cos B + \sin A \sin B$$

Substitute $A = 80^{\circ}$ and $B = 70^{\circ}$ to get:

$$\cos (80^{\circ} - 70^{\circ}) = \cos 80^{\circ} \cos 70^{\circ} + \sin 80^{\circ} \sin 70^{\circ}$$

Therefore, you can rewrite the expression as:

$$\cos 80^{\circ} \cos 70^{\circ} - \sin 80^{\circ} \sin 70^{\circ} = -\cos (80^{\circ} - 70^{\circ})$$

Q: How can I evaluate the expression $-\cos (80^{\circ} - 70^{\circ})$?

A: To evaluate the expression, you need to find the value of $\cos (80^{\circ} - 70^{\circ})$. Since $80^{\circ} - 70^{\circ} = 10^{\circ}$, you can substitute this value into the expression:

$$-\cos (80^{\circ} - 70^{\circ}) = -\cos 10^{\circ}$$

Using a calculator or a trigonometric table, you can find that:

$$\cos 10^{\circ} = \frac{\sqrt{3}}{2}$$

Therefore, the expression can be rewritten as:

$$-\cos (80^{\circ} - 70^{\circ}) = -\frac{\sqrt{3}}{2}$$

Q: What is the final answer to the expression $\cos 80^{\circ} \cos 70^{\circ} - \sin 80^{\circ} \sin 70^{\circ}$?

A: The final answer to the expression is:

$$\cos 80^{\circ} \cos 70^{\circ} - \sin 80^{\circ} \sin 70^{\circ} = -\frac{\sqrt{3}}{2}$$

Q: How can I apply this result in real-world applications?

A: This result can be used in various applications, including physics, engineering, and navigation. For example, in physics, you can use this result to calculate the cosine of the difference of two angles in a right triangle. In engineering, you can use this result to design and analyze mechanical systems that involve trigonometric functions. In navigation, you can use this result to calculate the position and orientation of a vehicle or a satellite.

Conclusion

In this article, we have answered some frequently asked questions about trigonometric expressions. We have provided step-by-step solutions to simplify and evaluate the expression $\cos 80^{\circ} \cos 70^{\circ} - \sin 80^{\circ} \sin 70^{\circ}$. We have also discussed the applications of this result in real-world scenarios.