Find the exact value of the quotient.\ \frac{\cos\left(\frac{\pi}{6}\right)}{\sin\left(\frac{\pi}{6}\right)}\ The exact value of \frac{\cos\left(\frac{\pi}{6}\right)}{\sin\left(\frac{\pi}{6}\right)} is
The Correct Answer and Explanation is:
To find the exact value of the quotient cos(π6)sin(π6)\frac{\cos\left(\frac{\pi}{6}\right)}{\sin\left(\frac{\pi}{6}\right)}sin(6π)cos(6π), we need to recall the exact trigonometric values of cos(π6)\cos\left(\frac{\pi}{6}\right)cos(6π) and sin(π6)\sin\left(\frac{\pi}{6}\right)sin(6π).
Step 1: Know the standard trigonometric values
From the unit circle or standard trigonometric identities, the values of cosine and sine at π6\frac{\pi}{6}6π (or 30 degrees) are:
- cos(π6)=32\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}cos(6π)=23
- sin(π6)=12\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}sin(6π)=21
Step 2: Set up the quotient
Now, we can substitute these values into the original expression:cos(π6)sin(π6)=3212\frac{\cos\left(\frac{\pi}{6}\right)}{\sin\left(\frac{\pi}{6}\right)} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}sin(6π)cos(6π)=2123
Step 3: Simplify the fraction
To simplify, divide the numerator by the denominator. Dividing by a fraction is the same as multiplying by its reciprocal. So:3212=32×21=3\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \frac{\sqrt{3}}{2} \times \frac{2}{1} = \sqrt{3}2123=23×12=3
Final Answer:
Thus, the exact value of the quotient is:3\boxed{\sqrt{3}}3
Explanation:
The key steps to solving this problem are recognizing the exact values of cos\coscos and sin\sinsin at π6\frac{\pi}{6}6π, and then simplifying the fraction by multiplying by the reciprocal of the denominator. This technique is useful for simplifying trigonometric expressions quickly.
