Find The Exact Value Of The Expression:$\sin \frac{\pi}{12} \cos \frac{2 \pi}{3} + \cos \frac{\pi}{12} \sin \frac{2 \pi}{3}$\sin \frac{\pi}{12} \cos \frac{2 \pi}{3} + \cos \frac{\pi}{12} \sin \frac{2 \pi}{3} = \square$(Simplify Your

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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 simplifying a specific trigonometric expression involving sine and cosine functions.

The Expression to Simplify

The given expression is:

sin⁑π12cos⁑2Ο€3+cos⁑π12sin⁑2Ο€3\sin \frac{\pi}{12} \cos \frac{2 \pi}{3} + \cos \frac{\pi}{12} \sin \frac{2 \pi}{3}

This expression involves the product of sine and cosine functions with different angles. 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 sine of a sum of two angles:

sin⁑(A+B)=sin⁑Acos⁑B+cos⁑Asin⁑B\sin (A + B) = \sin A \cos B + \cos A \sin B

By comparing this identity with the given expression, we can see that:

sin⁑π12cos⁑2Ο€3+cos⁑π12sin⁑2Ο€3=sin⁑(Ο€12+2Ο€3)\sin \frac{\pi}{12} \cos \frac{2 \pi}{3} + \cos \frac{\pi}{12} \sin \frac{2 \pi}{3} = \sin \left(\frac{\pi}{12} + \frac{2 \pi}{3}\right)

Evaluating the Angle

Now, we need to evaluate the angle inside the sine function:

Ο€12+2Ο€3=Ο€12+8Ο€12=9Ο€12=3Ο€4\frac{\pi}{12} + \frac{2 \pi}{3} = \frac{\pi}{12} + \frac{8 \pi}{12} = \frac{9 \pi}{12} = \frac{3 \pi}{4}

Simplifying the Expression

Using the value of the angle, we can simplify the expression:

sin⁑(Ο€12+2Ο€3)=sin⁑3Ο€4\sin \left(\frac{\pi}{12} + \frac{2 \pi}{3}\right) = \sin \frac{3 \pi}{4}

Finding the Exact Value

To find the exact value of sin⁑3Ο€4\sin \frac{3 \pi}{4}, we can use the unit circle or trigonometric identities. From the unit circle, we know that:

sin⁑3Ο€4=sin⁑(90βˆ˜βˆ’45∘)=cos⁑45∘\sin \frac{3 \pi}{4} = \sin \left(90^\circ - 45^\circ\right) = \cos 45^\circ

Using Trigonometric Identities

Using the trigonometric identity for the cosine of a 45-degree angle, we can find the exact value:

cos⁑45∘=12=22\cos 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}

Conclusion

In this article, we simplified a trigonometric expression involving sine and cosine functions. We used trigonometric identities to rewrite the expression and evaluated the angle inside the sine function. Finally, we found the exact value of the expression using the unit circle and trigonometric identities.

Final Answer

The exact value of the expression is:

sin⁑π12cos⁑2Ο€3+cos⁑π12sin⁑2Ο€3=22\sin \frac{\pi}{12} \cos \frac{2 \pi}{3} + \cos \frac{\pi}{12} \sin \frac{2 \pi}{3} = \boxed{\frac{\sqrt{2}}{2}}

Applications

This trigonometric expression has numerous applications in various fields, including:

  • Physics: In the study of wave motion and oscillations, trigonometric functions are used to describe the behavior of waves.
  • Engineering: In the design of electrical circuits and mechanical systems, trigonometric functions are used to analyze and optimize system performance.
  • Navigation: In navigation systems, trigonometric functions are used to calculate distances and angles between locations.

Tips and Tricks

When simplifying trigonometric expressions, remember to:

  • Use trigonometric identities: Familiarize yourself with common trigonometric identities and use them to rewrite expressions.
  • Evaluate angles: Evaluate the angles inside trigonometric functions to simplify expressions.
  • Use the unit circle: Use the unit circle to find exact values of trigonometric functions.

By following these tips and tricks, you can simplify complex trigonometric expressions and find their exact values.

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Introduction

In our previous article, we simplified a trigonometric expression involving sine and cosine functions. In this article, we will answer some frequently asked questions about trigonometric expressions and provide additional tips and tricks for simplifying them.

Q&A

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

  • Pythagorean identity: sin⁑2x+cos⁑2x=1\sin^2 x + \cos^2 x = 1
  • Sum and difference identities: sin⁑(A+B)=sin⁑Acos⁑B+cos⁑Asin⁑B\sin (A + B) = \sin A \cos B + \cos A \sin B and cos⁑(A+B)=cos⁑Acos⁑Bβˆ’sin⁑Asin⁑B\cos (A + B) = \cos A \cos B - \sin A \sin B
  • Double-angle identities: sin⁑2x=2sin⁑xcos⁑x\sin 2x = 2 \sin x \cos x and cos⁑2x=cos⁑2xβˆ’sin⁑2x\cos 2x = \cos^2 x - \sin^2 x

Q: How do I evaluate angles inside trigonometric functions?

A: To evaluate angles inside trigonometric functions, you can use the following steps:

  1. Simplify the expression: Simplify the expression inside the trigonometric function using algebraic manipulations.
  2. Use trigonometric identities: Use trigonometric identities to rewrite the expression in a more manageable form.
  3. Evaluate the angle: Evaluate the angle inside the trigonometric function using the unit circle or trigonometric identities.

Q: What is the unit circle, and how do I use it to find exact values of trigonometric functions?

A: The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. It is used to find exact values of trigonometric functions by relating the coordinates of points on the circle to the values of the trigonometric functions.

To use the unit circle to find exact values of trigonometric functions, follow these steps:

  1. Draw the unit circle: Draw the unit circle on a coordinate plane.
  2. Identify the angle: Identify the angle inside the trigonometric function.
  3. Find the coordinates: Find the coordinates of the point on the unit circle corresponding to the angle.
  4. Relate the coordinates to the trigonometric function: Relate the coordinates to the value of the trigonometric function.

Q: How do I simplify complex trigonometric expressions?

A: To simplify complex trigonometric expressions, follow these steps:

  1. Use trigonometric identities: Use trigonometric identities to rewrite the expression in a more manageable form.
  2. Simplify the expression: Simplify the expression using algebraic manipulations.
  3. Evaluate the angle: Evaluate the angle inside the trigonometric function using the unit circle or trigonometric identities.

Tips and Tricks

When simplifying trigonometric expressions, remember to:

  • Use trigonometric identities: Familiarize yourself with common trigonometric identities and use them to rewrite expressions.
  • Evaluate angles: Evaluate the angles inside trigonometric functions to simplify expressions.
  • Use the unit circle: Use the unit circle to find exact values of trigonometric functions.
  • Simplify expressions: Simplify expressions using algebraic manipulations.

By following these tips and tricks, you can simplify complex trigonometric expressions and find their exact values.

Conclusion

In this article, we answered some frequently asked questions about trigonometric expressions and provided additional tips and tricks for simplifying them. We hope that this article has been helpful in your understanding of trigonometric expressions and how to simplify them.

Final Answer

The final answer to the trigonometric expression is:

sin⁑π12cos⁑2Ο€3+cos⁑π12sin⁑2Ο€3=22\sin \frac{\pi}{12} \cos \frac{2 \pi}{3} + \cos \frac{\pi}{12} \sin \frac{2 \pi}{3} = \boxed{\frac{\sqrt{2}}{2}}

Applications

Trigonometric expressions have numerous applications in various fields, including:

  • Physics: In the study of wave motion and oscillations, trigonometric functions are used to describe the behavior of waves.
  • Engineering: In the design of electrical circuits and mechanical systems, trigonometric functions are used to analyze and optimize system performance.
  • Navigation: In navigation systems, trigonometric functions are used to calculate distances and angles between locations.

Further Reading

For further reading on trigonometric expressions, we recommend the following resources:

  • Textbooks: "Trigonometry" by Michael Corral and "Trigonometry: A Unit Circle Approach" by Charles P. McKeague
  • Online resources: Khan Academy's trigonometry course and Wolfram Alpha's trigonometry calculator
  • Practice problems: Trigonometry practice problems on Brilliant.org and Mathway's trigonometry practice problems