Calculate The Value Of:1. $\cos 75^{\circ}$2. $\sin 70^{\circ} \sin 10^{\circ} + \cos 10^{\circ} \cos 70^{\circ}$3. $\sin 50^{\circ} \cos 10^{\circ} + \cos 50^{\circ} \sin 10^{\circ}$Expand And Simplify:1. $\cos

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 three trigonometric calculations and expansions, focusing on the values of cosine and sine functions.

Calculating the Value of Cosine

1. $\cos 75^{\circ}$

To calculate the value of $\cos 75^{\circ}$, we can use the angle addition formula for cosine:

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

Let's rewrite $75^{\circ}$ as the sum of two angles: $60^{\circ}$ and $15^{\circ}$. We can then apply the angle addition formula:

$$\cos 75^{\circ} = \cos (60^{\circ} + 15^{\circ}) = \cos 60^{\circ} \cos 15^{\circ} - \sin 60^{\circ} \sin 15^{\circ}$$

Using the exact values of cosine and sine for $60^{\circ}$ and $15^{\circ}$, we get:

$$\cos 75^{\circ} = \frac{1}{2} \cos 15^{\circ} - \frac{\sqrt{3}}{2} \sin 15^{\circ}$$

To simplify this expression, we can use the half-angle formula for cosine:

$$\cos 15^{\circ} = \cos \left(\frac{30^{\circ}}{2}\right) = \sqrt{\frac{1 + \cos 30^{\circ}}{2}}$$

$$\sin 15^{\circ} = \sin \left(\frac{30^{\circ}}{2}\right) = \sqrt{\frac{1 - \cos 30^{\circ}}{2}}$$

Substituting these expressions into the previous equation, we get:

$$\cos 75^{\circ} = \frac{1}{2} \sqrt{\frac{1 + \cos 30^{\circ}}{2}} - \frac{\sqrt{3}}{2} \sqrt{\frac{1 - \cos 30^{\circ}}{2}}$$

Simplifying this expression further, we get:

$$\cos 75^{\circ} = \frac{1}{4} \sqrt{2 + \sqrt{3}} - \frac{\sqrt{3}}{4} \sqrt{2 - \sqrt{3}}$$

2. $\sin 70^{\circ} \sin 10^{\circ} + \cos 10^{\circ} \cos 70^{\circ}$

To calculate the value of $\sin 70^{\circ} \sin 10^{\circ} + \cos 10^{\circ} \cos 70^{\circ}$, we can use the angle addition formula for sine:

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

Let's rewrite $70^{\circ}$ as the sum of two angles: $60^{\circ}$ and $10^{\circ}$. We can then apply the angle addition formula:

$$\sin 70^{\circ} = \sin (60^{\circ} + 10^{\circ}) = \sin 60^{\circ} \cos 10^{\circ} + \cos 60^{\circ} \sin 10^{\circ}$$

Using the exact values of sine and cosine for $60^{\circ}$ and $10^{\circ}$, we get:

$$\sin 70^{\circ} = \frac{\sqrt{3}}{2} \cos 10^{\circ} + \frac{1}{2} \sin 10^{\circ}$$

Now, let's multiply this expression by $\sin 10^{\circ}$:

$$\sin 70^{\circ} \sin 10^{\circ} = \frac{\sqrt{3}}{2} \cos 10^{\circ} \sin 10^{\circ} + \frac{1}{2} \sin^{2} 10^{\circ}$$

Using the angle addition formula for sine, we can rewrite the first term as:

$$\frac{\sqrt{3}}{2} \cos 10^{\circ} \sin 10^{\circ} = \frac{\sqrt{3}}{4} \sin 20^{\circ}$$

Now, let's add the second term to this expression:

$$\sin 70^{\circ} \sin 10^{\circ} + \cos 10^{\circ} \cos 70^{\circ} = \frac{\sqrt{3}}{4} \sin 20^{\circ} + \frac{1}{2} \sin^{2} 10^{\circ} + \cos 10^{\circ} \cos 70^{\circ}$$

Using the angle addition formula for cosine, we can rewrite the last term as:

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

Now, let's substitute this expression into the previous equation:

$$\sin 70^{\circ} \sin 10^{\circ} + \cos 10^{\circ} \cos 70^{\circ} = \frac{\sqrt{3}}{4} \sin 20^{\circ} + \frac{1}{2} \sin^{2} 10^{\circ} + \cos 80^{\circ}$$

3. $\sin 50^{\circ} \cos 10^{\circ} + \cos 50^{\circ} \sin 10^{\circ}$

To calculate the value of $\sin 50^{\circ} \cos 10^{\circ} + \cos 50^{\circ} \sin 10^{\circ}$, we can use the angle addition formula for sine:

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

Let's rewrite $50^{\circ}$ as the sum of two angles: $40^{\circ}$ and $10^{\circ}$. We can then apply the angle addition formula:

$$\sin 50^{\circ} = \sin (40^{\circ} + 10^{\circ}) = \sin 40^{\circ} \cos 10^{\circ} + \cos 40^{\circ} \sin 10^{\circ}$$

Using the exact values of sine and cosine for $40^{\circ}$ and $10^{\circ}$, we get:

$$\sin 50^{\circ} = \frac{2}{3} \cos 10^{\circ} + \frac{\sqrt{10}}{6} \sin 10^{\circ}$$

Now, let's multiply this expression by $\cos 10^{\circ}$:

$$\sin 50^{\circ} \cos 10^{\circ} = \frac{2}{3} \cos^{2} 10^{\circ} + \frac{\sqrt{10}}{6} \sin 10^{\circ} \cos 10^{\circ}$$

Using the angle addition formula for sine, we can rewrite the second term as:

$$\frac{\sqrt{10}}{6} \sin 10^{\circ} \cos 10^{\circ} = \frac{\sqrt{10}}{12} \sin 20^{\circ}$$

Now, let's add the second term to this expression:

$$\sin 50^{\circ} \cos 10^{\circ} + \cos 50^{\circ} \sin 10^{\circ} = \frac{2}{3} \cos^{2} 10^{\circ} + \frac{\sqrt{10}}{12} \sin 20^{\circ} + \cos 50^{\circ} \sin 10^{\circ}$$

Using the angle addition formula for cosine, we can rewrite the last term as:

$$\cos 50^{\circ} \sin 10^{\circ} = \cos (50^{\circ} + 10^{\circ}) = \cos 60^{\circ}$$

Now, let's substitute this expression into the previous equation:

$$\sin 50^{\circ} \cos 10^{\circ} + \cos 50^{\circ} \sin 10^{\circ} = \frac{2}{3} \cos^{2} 10^{\circ} + \frac{\sqrt{10}}{12} \sin 20^{\circ} + \frac{1}{2}$$

Expanding and Simplifying

1. $\cos (A + B)$

To expand and simplify the expression $\cos (A + B)$, we can use the angle addition formula for cosine:

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

Let's rewrite this expression as:

\cos (A + B) = \cos A \cos B - \sin A<br/> **Trigonometric Calculations and Expansions: Q&A** ============================================= **Introduction** --------------- In our previous article, we explored three trigonometric calculations and expansions, focusing on the values of cosine and sine functions. In this article, we will answer some frequently asked questions related to these calculations and expansions. **Q: What is the angle addition formula for cosine?** ----------------------------------------------- A: The angle addition formula for cosine is: $\cos (a + b) = \cos a \cos b - \sin a \sin b

Q: How do I calculate the value of $\cos 75^{\circ}$?

A: To calculate the value of $\cos 75^{\circ}$, you can use the angle addition formula for cosine:

$$\cos 75^{\circ} = \cos (60^{\circ} + 15^{\circ}) = \cos 60^{\circ} \cos 15^{\circ} - \sin 60^{\circ} \sin 15^{\circ}$$

Q: What is the value of $\sin 70^{\circ} \sin 10^{\circ} + \cos 10^{\circ} \cos 70^{\circ}$?

A: To calculate the value of $\sin 70^{\circ} \sin 10^{\circ} + \cos 10^{\circ} \cos 70^{\circ}$, you can use the angle addition formula for sine:

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

Let's rewrite $70^{\circ}$ as the sum of two angles: $60^{\circ}$ and $10^{\circ}$. We can then apply the angle addition formula:

$$\sin 70^{\circ} = \sin (60^{\circ} + 10^{\circ}) = \sin 60^{\circ} \cos 10^{\circ} + \cos 60^{\circ} \sin 10^{\circ}$$

Using the exact values of sine and cosine for $60^{\circ}$ and $10^{\circ}$, we get:

$$\sin 70^{\circ} = \frac{\sqrt{3}}{2} \cos 10^{\circ} + \frac{1}{2} \sin 10^{\circ}$$

Now, let's multiply this expression by $\sin 10^{\circ}$:

$$\sin 70^{\circ} \sin 10^{\circ} = \frac{\sqrt{3}}{2} \cos 10^{\circ} \sin 10^{\circ} + \frac{1}{2} \sin^{2} 10^{\circ}$$

Using the angle addition formula for sine, we can rewrite the first term as:

$$\frac{\sqrt{3}}{2} \cos 10^{\circ} \sin 10^{\circ} = \frac{\sqrt{3}}{4} \sin 20^{\circ}$$

Now, let's add the second term to this expression:

$$\sin 70^{\circ} \sin 10^{\circ} + \cos 10^{\circ} \cos 70^{\circ} = \frac{\sqrt{3}}{4} \sin 20^{\circ} + \frac{1}{2} \sin^{2} 10^{\circ} + \cos 10^{\circ} \cos 70^{\circ}$$

Using the angle addition formula for cosine, we can rewrite the last term as:

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

Now, let's substitute this expression into the previous equation:

$$\sin 70^{\circ} \sin 10^{\circ} + \cos 10^{\circ} \cos 70^{\circ} = \frac{\sqrt{3}}{4} \sin 20^{\circ} + \frac{1}{2} \sin^{2} 10^{\circ} + \cos 80^{\circ}$$

Q: What is the value of $\sin 50^{\circ} \cos 10^{\circ} + \cos 50^{\circ} \sin 10^{\circ}$?

A: To calculate the value of $\sin 50^{\circ} \cos 10^{\circ} + \cos 50^{\circ} \sin 10^{\circ}$, you can use the angle addition formula for sine:

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

Let's rewrite $50^{\circ}$ as the sum of two angles: $40^{\circ}$ and $10^{\circ}$. We can then apply the angle addition formula:

$$\sin 50^{\circ} = \sin (40^{\circ} + 10^{\circ}) = \sin 40^{\circ} \cos 10^{\circ} + \cos 40^{\circ} \sin 10^{\circ}$$

Using the exact values of sine and cosine for $40^{\circ}$ and $10^{\circ}$, we get:

$$\sin 50^{\circ} = \frac{2}{3} \cos 10^{\circ} + \frac{\sqrt{10}}{6} \sin 10^{\circ}$$

Now, let's multiply this expression by $\cos 10^{\circ}$:

$$\sin 50^{\circ} \cos 10^{\circ} = \frac{2}{3} \cos^{2} 10^{\circ} + \frac{\sqrt{10}}{6} \sin 10^{\circ} \cos 10^{\circ}$$

Using the angle addition formula for sine, we can rewrite the second term as:

$$\frac{\sqrt{10}}{6} \sin 10^{\circ} \cos 10^{\circ} = \frac{\sqrt{10}}{12} \sin 20^{\circ}$$

Now, let's add the second term to this expression:

$$\sin 50^{\circ} \cos 10^{\circ} + \cos 50^{\circ} \sin 10^{\circ} = \frac{2}{3} \cos^{2} 10^{\circ} + \frac{\sqrt{10}}{12} \sin 20^{\circ} + \cos 50^{\circ} \sin 10^{\circ}$$

Using the angle addition formula for cosine, we can rewrite the last term as:

$$\cos 50^{\circ} \sin 10^{\circ} = \cos (50^{\circ} + 10^{\circ}) = \cos 60^{\circ}$$

Now, let's substitute this expression into the previous equation:

$$\sin 50^{\circ} \cos 10^{\circ} + \cos 50^{\circ} \sin 10^{\circ} = \frac{2}{3} \cos^{2} 10^{\circ} + \frac{\sqrt{10}}{12} \sin 20^{\circ} + \frac{1}{2}$$

Conclusion

In this article, we have answered some frequently asked questions related to trigonometric calculations and expansions. We have also provided step-by-step solutions to three trigonometric calculations and expansions, focusing on the values of cosine and sine functions. We hope that this article has been helpful in understanding these calculations and expansions.

Additional Resources

For more information on trigonometric calculations and expansions, please refer to the following resources:

• Trigonometry Wikipedia Article
• Trigonometric Calculations and Expansions
• Trigonometric Identities

We hope that this article has been helpful in understanding trigonometric calculations and expansions. If you have any further questions or need additional assistance, please don't hesitate to ask.