Find The Exact Value Of The Expression By Using Appropriate Identities. Do Not Use A Calculator.$\[ \sin 74^{\circ} \cos 29^{\circ} - \cos 74^{\circ} \sin 29^{\circ} \\]$\[ \sin 74^{\circ} \cos 29^{\circ} - \cos 74^{\circ} \sin 29^{\circ}

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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 trigonometric expressions using appropriate identities. We will use the given expression sin⁡74∘cos⁡29∘−cos⁡74∘sin⁡29∘\sin 74^{\circ} \cos 29^{\circ} - \cos 74^{\circ} \sin 29^{\circ} as an example to demonstrate the steps involved.

Understanding the Given Expression

The given expression is a difference of two products of sine and cosine functions. It can be written as:

sin⁡74∘cos⁡29∘−cos⁡74∘sin⁡29∘\sin 74^{\circ} \cos 29^{\circ} - \cos 74^{\circ} \sin 29^{\circ}

This expression can be simplified using the trigonometric identity for the sine of a difference of two angles.

Using the Trigonometric Identity for Sine of a Difference

The trigonometric identity for the sine of a difference of two angles is given by:

sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin (A - B) = \sin A \cos B - \cos A \sin B

We can rewrite the given expression using this identity:

sin⁡74∘cos⁡29∘−cos⁡74∘sin⁡29∘=sin⁡(74∘−29∘)\sin 74^{\circ} \cos 29^{\circ} - \cos 74^{\circ} \sin 29^{\circ} = \sin (74^{\circ} - 29^{\circ})

Simplifying the Expression

Now, we can simplify the expression by evaluating the sine of the difference of the two angles:

sin⁡(74∘−29∘)=sin⁡45∘\sin (74^{\circ} - 29^{\circ}) = \sin 45^{\circ}

Evaluating the Sine of 45 Degrees

The sine of 45 degrees is a well-known value in trigonometry. It is equal to:

sin⁡45∘=12\sin 45^{\circ} = \frac{1}{\sqrt{2}}

Conclusion

In this article, we have simplified the given trigonometric expression using the trigonometric identity for the sine of a difference of two angles. We have shown that the expression can be rewritten as the sine of the difference of the two angles, and then evaluated the sine of 45 degrees to obtain the final result. This example demonstrates the importance of using trigonometric identities to simplify complex expressions and evaluate trigonometric functions.

Common Trigonometric Identities

Here are some common trigonometric identities that are used to simplify expressions:

  • Sine of a difference of two angles: sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin (A - B) = \sin A \cos B - \cos A \sin B
  • Cosine of a difference of two angles: cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\cos (A - B) = \cos A \cos B + \sin A \sin B
  • 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
  • Cosine of a sum of two angles: cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos (A + B) = \cos A \cos B - \sin A \sin B

Tips for Simplifying Trigonometric Expressions

Here are some tips for simplifying trigonometric expressions:

  • Use trigonometric identities: Trigonometric identities are a powerful tool for simplifying expressions. Use them to rewrite the expression in a more manageable form.
  • Look for common factors: Look for common factors in the expression and factor them out.
  • Use algebraic manipulations: Use algebraic manipulations to simplify the expression.
  • Check for errors: Check the expression for errors and make sure that it is correct.

Practice Problems

Here are some practice problems to help you practice simplifying trigonometric expressions:

  • Simplify the expression: sin⁡32∘cos⁡57∘−cos⁡32∘sin⁡57∘\sin 32^{\circ} \cos 57^{\circ} - \cos 32^{\circ} \sin 57^{\circ}
  • Simplify the expression: cos⁡19∘cos⁡41∘+sin⁡19∘sin⁡41∘\cos 19^{\circ} \cos 41^{\circ} + \sin 19^{\circ} \sin 41^{\circ}
  • Simplify the expression: sin⁡67∘cos⁡13∘+cos⁡67∘sin⁡13∘\sin 67^{\circ} \cos 13^{\circ} + \cos 67^{\circ} \sin 13^{\circ}

Conclusion

Q: What is the difference between the sine and cosine functions?

A: The sine and cosine functions are two of the most basic trigonometric functions. The sine function represents the ratio of the length of the side opposite a given angle to the length of the hypotenuse in a right triangle, while the cosine function represents the ratio of the length of the side adjacent to a given angle to the length of the hypotenuse.

Q: How do I simplify a trigonometric expression using trigonometric identities?

A: To simplify a trigonometric expression using trigonometric identities, you need to identify the type of identity that can be applied to the expression. For example, if the expression contains the sine of a difference of two angles, you can use the trigonometric identity for the sine of a difference of two angles to simplify it.

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

A: The trigonometric identity for the sine of a difference of two angles is given by:

sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin (A - B) = \sin A \cos B - \cos A \sin B

Q: How do I evaluate the sine of a specific angle?

A: To evaluate the sine of a specific angle, you need to use a trigonometric table or a calculator. The sine of an angle is a value between -1 and 1, and it represents the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle.

Q: What is the difference between the sine and cosine of complementary angles?

A: The sine and cosine of complementary angles are equal. For example, if the sine of an angle is 0.5, then the cosine of its complementary angle is also 0.5.

Q: How do I simplify a trigonometric expression using algebraic manipulations?

A: To simplify a trigonometric expression using algebraic manipulations, you need to use algebraic rules such as factoring, combining like terms, and canceling out common factors.

Q: What are some common trigonometric identities that I should know?

A: Some common trigonometric identities that you should know include:

  • Sine of a difference of two angles: sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin (A - B) = \sin A \cos B - \cos A \sin B
  • Cosine of a difference of two angles: cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\cos (A - B) = \cos A \cos B + \sin A \sin B
  • 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
  • Cosine of a sum of two angles: cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos (A + B) = \cos A \cos B - \sin A \sin B

Q: How do I practice simplifying trigonometric expressions?

A: To practice simplifying trigonometric expressions, you can try solving practice problems or working on exercises that involve simplifying trigonometric expressions. You can also use online resources or trigonometry textbooks to find practice problems and exercises.

Q: What are some real-world applications of trigonometry?

A: Trigonometry has numerous real-world applications, including:

  • Navigation: Trigonometry is used in navigation to calculate distances and directions between two points.
  • Physics: Trigonometry is used in physics to describe the motion of objects and the behavior of waves.
  • Engineering: Trigonometry is used in engineering to design and build structures such as bridges and buildings.
  • Computer Science: Trigonometry is used in computer science to create 3D graphics and animations.

Conclusion

In this article, we have answered some frequently asked questions about trigonometry and simplifying trigonometric expressions. We have provided explanations and examples to help you understand the concepts and techniques involved in simplifying trigonometric expressions. We have also provided some common trigonometric identities and real-world applications of trigonometry. Finally, we have included some practice problems to help you practice simplifying trigonometric expressions.