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The problem asks to show that the expression:

\[
20 \cos 30^\circ + 4 \sin 60^\circ - 2 \tan 60^\circ
\]

can be written in the form \( v \sqrt{k} \), where \( k \) is an integer.

### Step-by-Step Solution:

#### Step 1: Use standard trigonometric values

1. \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)
2. \( \sin 60^\circ = \frac{\sqrt{3}}{2} \)
3. \( \tan 60^\circ = \sqrt{3} \)

#### Step 2: Substitute these values into the expression

\[
20 \cos 30^\circ = 20 \times \frac{\sqrt{3}}{2} = 10 \sqrt{3}
\]
\[
4 \sin 60^\circ = 4 \times \frac{\sqrt{3}}{2} = 2 \sqrt{3}
\]
\[
2 \tan 60^\circ = 2 \times \sqrt{3} = 2 \sqrt{3}
\]

#### Step 3: Combine the terms

Now, the expression becomes:

\[
10 \sqrt{3} + 2 \sqrt{3} - 2 \sqrt{3}
\]

Simplifying:

\[
(10 \sqrt{3} + 2 \sqrt{3} - 2 \sqrt{3}) = 10 \sqrt{3}
\]

#### Step 4: Write in the desired form

We can see that the final result is \( 10 \sqrt{3} \), which is in the form \( v \sqrt{k} \), where \( v = 10 \) and \( k = 3 \).

Thus, the expression can be written as \( 10 \sqrt{3} \), where \( k = 3 \) is an integer.

### Conclusion:

The expression can indeed be written in the form \( v \sqrt{k} \), with \( v = 10 \) and \( k = 3 \).

---

Would you like a deeper explanation of any step? Here are five related questions for further exploration:

1. What are the values of common trigonometric functions for angles like 30°, 45°, and 60°?
2. How do you simplify expressions involving square roots in trigonometry?
3. Why do we use radians in trigonometry instead of degrees in advanced mathematics?
4. Can trigonometric functions of other special angles, such as 45°, also yield results involving square roots?
5. How does the use of identities simplify more complex trigonometric expressions?

**Tip:** Memorizing the values of basic trigonometric functions for standard angles (like 30°, 45°, and 60°) is extremely helpful in solving problems quickly.

Show that 20 cos 30° + 4 sin 60° - 2 tan 60° can be written in the form v√k where k is an integer.

Learn how to simplify the trigonometric expression 20 cos 30° + 4 sin 60° - 2 tan 60° and show that it can be written in the form v√k, where k is an integer. Includes step-by-step explanations for simplifying trigonometric expressions.

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