Verify The Identity.$\sin \left(x+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}(\sin X+\cos X$\]

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Verify the Identity: sin⁡(x+π4)=22(sin⁡x+cos⁡x\sin \left(x+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}(\sin x+\cos x

In mathematics, trigonometric identities are essential for solving problems and simplifying expressions. One of the most common trigonometric identities is the angle addition formula, which states that sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin (A+B) = \sin A \cos B + \cos A \sin B. In this article, we will verify the identity sin⁡(x+π4)=22(sin⁡x+cos⁡x)\sin \left(x+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}(\sin x+\cos x) using the angle addition formula.

The Angle Addition Formula

The angle addition formula is a fundamental concept in trigonometry, and it is used to find the sine and cosine of the sum of two angles. The formula states that:

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

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

Verifying the Identity

To verify the identity sin⁡(x+π4)=22(sin⁡x+cos⁡x)\sin \left(x+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}(\sin x+\cos x), we will use the angle addition formula. We will start by substituting A=xA = x and B=π4B = \frac{\pi}{4} into the angle addition formula.

sin⁡(x+π4)=sin⁡xcos⁡π4+cos⁡xsin⁡π4\sin \left(x+\frac{\pi}{4}\right) = \sin x \cos \frac{\pi}{4} + \cos x \sin \frac{\pi}{4}

Evaluating the Trigonometric Functions

Now, we need to evaluate the trigonometric functions cos⁡π4\cos \frac{\pi}{4} and sin⁡π4\sin \frac{\pi}{4}. We know that cos⁡π4=22\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} and sin⁡π4=22\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}.

sin⁡(x+π4)=sin⁡x22+cos⁡x22\sin \left(x+\frac{\pi}{4}\right) = \sin x \frac{\sqrt{2}}{2} + \cos x \frac{\sqrt{2}}{2}

Simplifying the Expression

Now, we can simplify the expression by combining like terms.

sin⁡(x+π4)=22(sin⁡x+cos⁡x)\sin \left(x+\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}(\sin x + \cos x)

In this article, we verified the identity sin⁡(x+π4)=22(sin⁡x+cos⁡x)\sin \left(x+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}(\sin x+\cos x) using the angle addition formula. We started by substituting A=xA = x and B=π4B = \frac{\pi}{4} into the angle addition formula, and then evaluated the trigonometric functions cos⁡π4\cos \frac{\pi}{4} and sin⁡π4\sin \frac{\pi}{4}. Finally, we simplified the expression by combining like terms.

The Importance of Trigonometric Identities

Trigonometric identities are essential for solving problems and simplifying expressions in mathematics. They provide a way to express complex trigonometric functions in terms of simpler functions, making it easier to solve problems and understand mathematical concepts. In this article, we verified the identity sin⁡(x+π4)=22(sin⁡x+cos⁡x)\sin \left(x+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}(\sin x+\cos x) using the angle addition formula, demonstrating the importance of trigonometric identities in mathematics.

Common Applications of Trigonometric Identities

Trigonometric identities have many common applications in mathematics and science. They are used to solve problems in trigonometry, calculus, and physics, and are essential for understanding many mathematical concepts. Some common applications of trigonometric identities include:

  • Solving trigonometric equations: Trigonometric identities are used to solve trigonometric equations, which are essential for understanding many mathematical concepts.
  • Simplifying trigonometric expressions: Trigonometric identities are used to simplify trigonometric expressions, making it easier to solve problems and understand mathematical concepts.
  • Understanding periodic functions: Trigonometric identities are used to understand periodic functions, which are essential for understanding many mathematical concepts.
  • Solving problems in physics and engineering: Trigonometric identities are used to solve problems in physics and engineering, such as calculating distances and angles.

Q: What is the angle addition formula?

A: The angle addition formula is a fundamental concept in trigonometry, which states that 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⁡B−sin⁡Asin⁡B\cos (A+B) = \cos A \cos B - \sin A \sin B.

Q: How do I verify the identity sin⁡(x+π4)=22(sin⁡x+cos⁡x)\sin \left(x+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}(\sin x+\cos x)?

A: To verify the identity, you can use the angle addition formula by substituting A=xA = x and B=π4B = \frac{\pi}{4} into the formula. Then, evaluate the trigonometric functions cos⁡π4\cos \frac{\pi}{4} and sin⁡π4\sin \frac{\pi}{4}, and simplify the expression by combining like terms.

Q: What are the values of cos⁡π4\cos \frac{\pi}{4} and sin⁡π4\sin \frac{\pi}{4}?

A: The values of cos⁡π4\cos \frac{\pi}{4} and sin⁡π4\sin \frac{\pi}{4} are both 22\frac{\sqrt{2}}{2}.

Q: How do I simplify the expression sin⁡(x+π4)=sin⁡x22+cos⁡x22\sin \left(x+\frac{\pi}{4}\right) = \sin x \frac{\sqrt{2}}{2} + \cos x \frac{\sqrt{2}}{2}?

A: To simplify the expression, you can combine like terms by adding the two terms together.

Q: What is the final simplified expression for sin⁡(x+π4)\sin \left(x+\frac{\pi}{4}\right)?

A: The final simplified expression for sin⁡(x+π4)\sin \left(x+\frac{\pi}{4}\right) is 22(sin⁡x+cos⁡x)\frac{\sqrt{2}}{2}(\sin x + \cos x).

Q: Why are trigonometric identities important in mathematics?

A: Trigonometric identities are important in mathematics because they provide a way to express complex trigonometric functions in terms of simpler functions, making it easier to solve problems and understand mathematical concepts.

Q: What are some common applications of trigonometric identities?

A: Some common applications of trigonometric identities include solving trigonometric equations, simplifying trigonometric expressions, understanding periodic functions, and solving problems in physics and engineering.

Q: Can you provide more examples of trigonometric identities?

A: Yes, here are a few more examples of trigonometric identities:

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

Q: How do I use trigonometric identities to solve problems?

A: To use trigonometric identities to solve problems, you can start by identifying the type of problem you are trying to solve. Then, use the appropriate trigonometric identity to simplify the expression and solve the problem.

Q: What are some tips for verifying trigonometric identities?

A: Some tips for verifying trigonometric identities include:

  • Start by identifying the type of problem you are trying to solve.
  • Use the appropriate trigonometric identity to simplify the expression.
  • Evaluate the trigonometric functions and simplify the expression by combining like terms.
  • Check your work by plugging in values for the variables and verifying that the expression is true.