Final answer:
The expression '12 cos 30° + 2 tan 60°' simplifies to '8 √3', which can be further expressed as √192, confirming that it can be written in the form of √K, where K is the integer 192.
Step-by-step explanation:
The question asks to show that '12 cos 30° + 2 tan 60°' can be written in the form of √K where K is an integer. To solve this, we'll use trigonometric identities and the properties of square roots.
First, let's simplify the expression:
12 cos 30° is equivalent to 12 √3 / 2, because cos 30° equals √3 / 2.
That simplifies to 6 √3.
Next, 2 tan 60° is equivalent to 2 √3, because tan 60° equals √3.
Adding both terms, we get 6 √3 + 2 √3, which simplifies to 8 √3.Now, notice that '8 √3' can be written as √(64 × 3) since 8 squared is 64 and √3 squared is 3. Therefore, the expression can be written as √192, where 192 is an integer.
Thus, √K is √192 with K being 192.