Express Sin ⁡ 4 X \sin 4x Sin 4 X In Terms Of X X X .

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Introduction

In trigonometry, expressing a trigonometric function in terms of another is a fundamental concept. One such problem is to express $\sin 4x$ in terms of $x$. This involves using trigonometric identities to simplify the given expression. In this article, we will explore the steps to express $\sin 4x$ in terms of $x$.

Using the Double Angle Formula

The double angle formula for sine is given by:

$$\sin 2\theta = 2\sin \theta \cos \theta$$

We can use this formula to express $\sin 4x$ in terms of $x$. Let's start by substituting $2x$ for $\theta$ in the double angle formula:

$$\sin 4x = \sin 2(2x)$$

Using the double angle formula, we get:

$$\sin 4x = 2\sin 2x \cos 2x$$

Now, we need to express $\sin 2x$ and $\cos 2x$ in terms of $x$. We can use the double angle formula again to get:

$$\sin 2x = 2\sin x \cos x$$

$$\cos 2x = 1 - 2\sin^2 x$$

Substituting these expressions into the previous equation, we get:

$$\sin 4x = 2(2\sin x \cos x)(1 - 2\sin^2 x)$$

Simplifying this expression, we get:

$$\sin 4x = 4\sin x \cos x - 4\sin^3 x$$

Using the Triple Angle Formula

Another way to express $\sin 4x$ in terms of $x$ is to use the triple angle formula for sine:

$$\sin 3\theta = 3\sin \theta - 4\sin^3 \theta$$

Let's substitute $2x$ for $\theta$ in this formula:

$$\sin 6x = 3\sin 2x - 4\sin^3 2x$$

Using the double angle formula, we get:

$$\sin 6x = 3(2\sin x \cos x) - 4(2\sin x \cos x)^3$$

Simplifying this expression, we get:

$$\sin 6x = 6\sin x \cos x - 32\sin^3 x \cos^3 x$$

Now, we can use the identity $\sin 4x = \sin (6x - 2x)$ to express $\sin 4x$ in terms of $x$:

$$\sin 4x = \sin (6x - 2x)$$

Using the sum and difference formula for sine, we get:

$$\sin 4x = \sin 6x \cos 2x - \cos 6x \sin 2x$$

Substituting the expressions for $\sin 6x$ and $\cos 6x$ in terms of $x$, we get:

$$\sin 4x = (6\sin x \cos x - 32\sin^3 x \cos^3 x)\cos 2x - (1 - 32\sin^2 x \cos^2 x)\sin 2x$$

Simplifying this expression, we get:

$$\sin 4x = 6\sin x \cos x \cos 2x - 32\sin^3 x \cos^3 x \cos 2x - \sin 2x + 32\sin^2 x \cos^2 x \sin 2x$$

Conclusion

In this article, we have explored two methods to express $\sin 4x$ in terms of $x$. The first method uses the double angle formula, while the second method uses the triple angle formula. Both methods involve using trigonometric identities to simplify the given expression. The final expression for $\sin 4x$ in terms of $x$ is:

$$\sin 4x = 4\sin x \cos x - 4\sin^3 x$$

or

$$\sin 4x = 6\sin x \cos x \cos 2x - 32\sin^3 x \cos^3 x \cos 2x - \sin 2x + 32\sin^2 x \cos^2 x \sin 2x$$

These expressions can be used to simplify trigonometric expressions involving $\sin 4x$.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Trigonometry" by I. M. Gelfand and M. L. Gelfand

Glossary

• Double angle formula: A trigonometric identity that relates the sine and cosine of a double angle to the sine and cosine of the original angle.
• Triple angle formula: A trigonometric identity that relates the sine of a triple angle to the sine and cosine of the original angle.
• Sum and difference formula: A trigonometric identity that relates the sine and cosine of a sum or difference of two angles to the sine and cosine of the individual angles.
  Q&A: Expressing $\sin 4x$ in Terms of $x$ =============================================

Frequently Asked Questions

Q: What is the double angle formula for sine?

A: The double angle formula for sine is given by:

$$\sin 2\theta = 2\sin \theta \cos \theta$$

Q: How can I use the double angle formula to express $\sin 4x$ in terms of $x$?

A: To use the double angle formula to express $\sin 4x$ in terms of $x$, you can substitute $2x$ for $\theta$ in the formula:

$$\sin 4x = \sin 2(2x)$$

Using the double angle formula, you get:

$$\sin 4x = 2\sin 2x \cos 2x$$

Q: What is the triple angle formula for sine?

A: The triple angle formula for sine is given by:

$$\sin 3\theta = 3\sin \theta - 4\sin^3 \theta$$

Q: How can I use the triple angle formula to express $\sin 4x$ in terms of $x$?

A: To use the triple angle formula to express $\sin 4x$ in terms of $x$, you can substitute $2x$ for $\theta$ in the formula:

$$\sin 6x = 3\sin 2x - 4\sin^3 2x$$

Using the double angle formula, you get:

$$\sin 6x = 3(2\sin x \cos x) - 4(2\sin x \cos x)^3$$

Simplifying this expression, you get:

$$\sin 6x = 6\sin x \cos x - 32\sin^3 x \cos^3 x$$

Q: What is the sum and difference formula for sine?

A: The sum and difference formula for sine is given by:

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

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

Q: How can I use the sum and difference formula to express $\sin 4x$ in terms of $x$?

A: To use the sum and difference formula to express $\sin 4x$ in terms of $x$, you can substitute $6x$ for $A$ and $2x$ for $B$ in the formula:

$$\sin (6x - 2x) = \sin 6x \cos 2x - \cos 6x \sin 2x$$

Using the expressions for $\sin 6x$ and $\cos 6x$ in terms of $x$, you get:

$$\sin 4x = (6\sin x \cos x - 32\sin^3 x \cos^3 x)\cos 2x - (1 - 32\sin^2 x \cos^2 x)\sin 2x$$

Q: What is the final expression for $\sin 4x$ in terms of $x$?

A: The final expression for $\sin 4x$ in terms of $x$ is:

$$\sin 4x = 4\sin x \cos x - 4\sin^3 x$$

or

$$\sin 4x = 6\sin x \cos x \cos 2x - 32\sin^3 x \cos^3 x \cos 2x - \sin 2x + 32\sin^2 x \cos^2 x \sin 2x$$

Q: How can I simplify trigonometric expressions involving $\sin 4x$?

A: You can simplify trigonometric expressions involving $\sin 4x$ by using the final expressions for $\sin 4x$ in terms of $x$.

Q: What are some common mistakes to avoid when expressing $\sin 4x$ in terms of $x$?

A: Some common mistakes to avoid when expressing $\sin 4x$ in terms of $x$ include:

• Not using the correct trigonometric identities
• Not simplifying the expressions correctly
• Not checking the final expressions for errors

Q: How can I practice expressing $\sin 4x$ in terms of $x$?

A: You can practice expressing $\sin 4x$ in terms of $x$ by working through examples and exercises. You can also try using different trigonometric identities and formulas to see if you can come up with alternative expressions for $\sin 4x$.

Conclusion

Expressing $\sin 4x$ in terms of $x$ is a fundamental concept in trigonometry. By using the double angle formula, triple angle formula, and sum and difference formula, you can simplify trigonometric expressions involving $\sin 4x$. Remember to check your final expressions for errors and to practice expressing $\sin 4x$ in terms of $x$ to become more confident in your skills.