Find the exact value of the expression. Do not use a calculator.

4 cosine 30 degrees minus 5 sine 30 degrees4cos30°−5sin30°

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Part 1

4 cosine 30 degrees minus 5 sine 30 degrees4cos30°−5sin30°equals=enter your response here

​(Simplify your answer. Type an exact​ answer, using radicals as needed. Use integers or fractions for any numbers in the​ expression.)

To find the exact value of the expression \(4 \cos 30^\circ - 5 \sin 30^\circ\), let's go through each part step-by-step.

### Step 1: Substitute the exact values for \(\cos 30^\circ\) and \(\sin 30^\circ\)
The exact values for these trigonometric functions are:
\[
\cos 30^\circ = \frac{\sqrt{3}}{2}
\]
\[
\sin 30^\circ = \frac{1}{2}
\]

### Step 2: Substitute these values into the expression
The expression is:
\[
4 \cos 30^\circ - 5 \sin 30^\circ
\]
Substitute \(\cos 30^\circ = \frac{\sqrt{3}}{2}\) and \(\sin 30^\circ = \frac{1}{2}\):
\[
= 4 \cdot \frac{\sqrt{3}}{2} - 5 \cdot \frac{1}{2}
\]

### Step 3: Simplify each term
Calculate each term separately:
\[
4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3}
\]
\[
5 \cdot \frac{1}{2} = \frac{5}{2}
\]

### Step 4: Combine the terms
Now, the expression simplifies to:
\[
2\sqrt{3} - \frac{5}{2}
\]

### Final Answer
The exact value of the expression is:
\[
2\sqrt{3} - \frac{5}{2}
\]

Would you like further details on these steps or have any questions?

Here are some related questions:
1. What are the exact values for other common trigonometric angles?
2. How would this expression change if we used radians instead of degrees?
3. How do we simplify expressions involving radicals and fractions?
4. What if \(\sin 30^\circ\) was negative; how would that affect the answer?
5. Can we represent this result in decimal form without a calculator?

**Tip:** Memorizing common trigonometric values (like for \(30^\circ\), \(45^\circ\), \(60^\circ\)) can simplify solving these types of problems.

Find the exact value of the expression: 4cos30° - 5sin30°. (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)

Solve the trigonometric expression 4cos30° - 5sin30° by finding exact values for sin and cos at 30° and simplifying the result. Follow step-by-step calculations to get the precise answer using radicals. Perfect for high school trigonometry students.

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