Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Sep 02, 2023

Find the exact value of $6 \cos \frac{\pi}{3} - 3 \tan \frac{\pi}{6}$ without using a calculator.

STEP 1

Assumptions1. We are working in the realm of trigonometry. . We are using radians, not degrees.
3. We are familiar with the unit circle and the values of sine, cosine, and tangent at common angles.

STEP 2

We need to find the exact values of cosine and tangent at the given angles. From the unit circle, we know that $\cos \frac{\pi}{} = \frac{1}{2}$ $\tan \frac{\pi}{6} = \frac{\sqrt{}}{}$

STEP 3

Now, we substitute these values into the original expression.
$6 \cos \frac{\pi}{3}-3 \tan \frac{\pi}{6} =6 \cdot \frac{1}{2} -3 \cdot \frac{\sqrt{3}}{3}$

STEP 4

implify the expression by performing the multiplications.
$6 \cdot \frac{1}{2} -3 \cdot \frac{\sqrt{3}}{3} =3 - \sqrt{3}$ So, the exact value of the expression $6 \cos \frac{\pi}{3}-3 \tan \frac{\pi}{6}$ is $3 - \sqrt{3}$ .

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