v0​sin(α)⋅v0​cos(α)x​ how does it equal x*tan(a)

The expression \( \frac{v_0 \sin(\alpha)}{v_0 \cos(\alpha)} \times x \) simplifies to \( x \tan(\alpha) \) based on trigonometric identities. Let's break down the reasoning step by step.

### Expression Breakdown
Consider the expression:

\[
\frac{v_0 \sin(\alpha)}{v_0 \cos(\alpha)} \times x
\]

### Step 1: Simplify the Fraction
First, simplify the fraction inside the expression:

\[
\frac{v_0 \sin(\alpha)}{v_0 \cos(\alpha)}
\]

Since \( v_0 \) is present in both the numerator and denominator, it cancels out:

\[
\frac{v_0 \sin(\alpha)}{v_0 \cos(\alpha)} = \frac{\sin(\alpha)}{\cos(\alpha)}
\]

### Step 2: Apply the Trigonometric Identity
The fraction \( \frac{\sin(\alpha)}{\cos(\alpha)} \) is a well-known trigonometric identity for the tangent function:

\[
\frac{\sin(\alpha)}{\cos(\alpha)} = \tan(\alpha)
\]

### Step 3: Multiply by \( x \)
Now, multiplying by \( x \):

\[
\frac{\sin(\alpha)}{\cos(\alpha)} \times x = \tan(\alpha) \times x
\]

This simplifies to:

\[
x \tan(\alpha)
\]

### Conclusion
Thus:

\[
\frac{v_0 \sin(\alpha)}{v_0 \cos(\alpha)} \times x = x \tan(\alpha)
\]

This is why \( \frac{v_0 \sin(\alpha)}{v_0 \cos(\alpha)} \times x \) simplifies to \( x \tan(\alpha) \).

Would you like to explore more about trigonometric identities or any other related topic?

Here are five related questions you might consider:
1. How is the tangent function used in other areas of physics?
2. What are other important trigonometric identities and their applications?
3. Can the tangent function be derived geometrically using a right triangle?
4. How does the tangent of an angle relate to slope in coordinate geometry?
5. What is the role of trigonometry in solving projectile motion problems?

**Tip:** Understanding the key trigonometric identities like \( \tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)} \) is crucial, as they frequently appear in physics and engineering problems.

Learn how to simplify the expression \( \frac{v_0 \sin(\alpha)}{v_0 \cos(\alpha)} \times x \) using trigonometric identities. This problem demonstrates the application of the tangent function and is suitable for high school students studying trigonometry.

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