Use A Sum Or Difference Identity To Find The Exact Value Of Sin ⁡ 75 ∘ \sin 75^{\circ} Sin 7 5 ∘ .

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Introduction

In trigonometry, sum and difference identities are used to simplify and evaluate trigonometric expressions. These identities are essential in solving problems involving trigonometric functions, especially when dealing with angles that are not standard angles. In this article, we will use a sum or difference identity to find the exact value of $\sin 75^{\circ}$.

Understanding Sum and Difference Identities

Sum and difference identities are used to express the sine and cosine of the sum or difference of two angles in terms of the sines and cosines of the individual angles. The sum and difference identities for sine and cosine are given by:

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

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

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

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

Finding the Exact Value of $\sin 75^{\circ}$

To find the exact value of $\sin 75^{\circ}$, we can use the sum identity for sine:

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

We can rewrite $75^{\circ}$ as the sum of two standard angles: $45^{\circ}$ and $30^{\circ}$. Therefore, we can write:

$\sin 75^{\circ} = \sin (45^{\circ} + 30^{\circ})$

Using the sum identity for sine, we get:

$\sin 75^{\circ} = \sin 45^{\circ} \cos 30^{\circ} + \cos 45^{\circ} \sin 30^{\circ}$

Evaluating the Trigonometric Functions

To evaluate the trigonometric functions, we need to know their exact values. The exact values of the trigonometric functions for standard angles are given by:

$\sin 45^{\circ} = \frac{\sqrt{2}}{2}$

$\cos 45^{\circ} = \frac{\sqrt{2}}{2}$

$\sin 30^{\circ} = \frac{1}{2}$

$\cos 30^{\circ} = \frac{\sqrt{3}}{2}$

Substituting the Exact Values

Substituting the exact values of the trigonometric functions, we get:

$\sin 75^{\circ} = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$

Simplifying the Expression

Simplifying the expression, we get:

$\sin 75^{\circ} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}$

Combining the Terms

Combining the terms, we get:

$\sin 75^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4}$

Conclusion

In this article, we used a sum identity to find the exact value of $\sin 75^{\circ}$. We started by rewriting $75^{\circ}$ as the sum of two standard angles: $45^{\circ}$ and $30^{\circ}$. We then used the sum identity for sine to express $\sin 75^{\circ}$ in terms of the sines and cosines of the individual angles. Finally, we evaluated the trigonometric functions and simplified the expression to find the exact value of $\sin 75^{\circ}$.

Final Answer

The final answer is $\boxed{\frac{\sqrt{6} + \sqrt{2}}{4}}$.

Additional Examples

Here are some additional examples of using sum and difference identities to find the exact values of trigonometric functions:

• $\sin (90^{\circ} - x) = \cos x$
• $\cos (90^{\circ} - x) = \sin x$
• $\sin (A - B) = \sin A \cos B - \cos A \sin B$
• $\cos (A - B) = \cos A \cos B + \sin A \sin B$

These identities can be used to simplify and evaluate trigonometric expressions, especially when dealing with angles that are not standard angles.

Common Mistakes

Here are some common mistakes to avoid when using sum and difference identities:

• Not rewriting the angle as the sum or difference of two standard angles
• Not using the correct identity for the given angle
• Not evaluating the trigonometric functions correctly
• Not simplifying the expression correctly

By avoiding these common mistakes, you can ensure that you are using sum and difference identities correctly and finding the exact values of trigonometric functions.

Real-World Applications

Sum and difference identities have many real-world applications in fields such as engineering, physics, and computer science. For example:

• In engineering, sum and difference identities are used to design and analyze electrical circuits and mechanical systems.
• In physics, sum and difference identities are used to describe the motion of objects and the behavior of waves.
• In computer science, sum and difference identities are used to develop algorithms and solve problems involving trigonometric functions.

By understanding and applying sum and difference identities, you can solve problems and make decisions in a variety of fields.

Conclusion

In conclusion, sum and difference identities are essential tools in trigonometry and have many real-world applications. By understanding and applying these identities, you can find the exact values of trigonometric functions and solve problems in a variety of fields. Remember to rewrite the angle as the sum or difference of two standard angles, use the correct identity for the given angle, evaluate the trigonometric functions correctly, and simplify the expression correctly. With practice and experience, you can become proficient in using sum and difference identities to find the exact values of trigonometric functions.

Introduction

In our previous article, we discussed how to use sum and difference identities to find the exact values of trigonometric functions. In this article, we will answer some frequently asked questions about sum and difference identities in trigonometry.

Q: What are sum and difference identities in trigonometry?

A: Sum and difference identities are formulas that express the sine and cosine of the sum or difference of two angles in terms of the sines and cosines of the individual angles.

Q: What are the sum and difference identities for sine and cosine?

A: The sum and difference identities for sine and cosine are given by:

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

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

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

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

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

A: To use sum and difference identities to find the exact value of a trigonometric function, follow these steps:

1. Rewrite the angle as the sum or difference of two standard angles.
2. Use the correct identity for the given angle.
3. Evaluate the trigonometric functions correctly.
4. Simplify the expression correctly.

Q: What are some common mistakes to avoid when using sum and difference identities?

A: Some common mistakes to avoid when using sum and difference identities include:

• Not rewriting the angle as the sum or difference of two standard angles.
• Not using the correct identity for the given angle.
• Not evaluating the trigonometric functions correctly.
• Not simplifying the expression correctly.

Q: How do I apply sum and difference identities in real-world problems?

A: Sum and difference identities have many real-world applications in fields such as engineering, physics, and computer science. To apply sum and difference identities in real-world problems, follow these steps:

1. Identify the problem and the relevant trigonometric functions.
2. Rewrite the angle as the sum or difference of two standard angles.
3. Use the correct identity for the given angle.
4. Evaluate the trigonometric functions correctly.
5. Simplify the expression correctly.

Q: What are some examples of using sum and difference identities in real-world problems?

A: Some examples of using sum and difference identities in real-world problems include:

• Designing and analyzing electrical circuits and mechanical systems in engineering.
• Describing the motion of objects and the behavior of waves in physics.
• Developing algorithms and solving problems involving trigonometric functions in computer science.

Q: How do I practice using sum and difference identities?

A: To practice using sum and difference identities, follow these steps:

1. Start with simple problems and gradually move on to more complex ones.
2. Practice rewriting angles as the sum or difference of two standard angles.
3. Practice using the correct identity for the given angle.
4. Practice evaluating trigonometric functions correctly.
5. Practice simplifying expressions correctly.

Q: What are some resources for learning more about sum and difference identities?

A: Some resources for learning more about sum and difference identities include:

• Textbooks and online resources on trigonometry and mathematics.
• Online tutorials and video lectures on trigonometry and mathematics.
• Practice problems and exercises on trigonometry and mathematics.

Conclusion

In conclusion, sum and difference identities are essential tools in trigonometry and have many real-world applications. By understanding and applying these identities, you can find the exact values of trigonometric functions and solve problems in a variety of fields. Remember to rewrite the angle as the sum or difference of two standard angles, use the correct identity for the given angle, evaluate the trigonometric functions correctly, and simplify the expression correctly. With practice and experience, you can become proficient in using sum and difference identities to find the exact values of trigonometric functions.

Final Answer

The final answer is $\boxed{\frac{\sqrt{6} + \sqrt{2}}{4}}$.

Additional Resources

Here are some additional resources for learning more about sum and difference identities:

• Khan Academy: Trigonometry
• MIT OpenCourseWare: Trigonometry
• Wolfram Alpha: Trigonometry
• Mathway: Trigonometry

These resources can help you learn more about sum and difference identities and practice using them to find the exact values of trigonometric functions.