Question

5.3
score: 9/22 answered: 9/22
question 10
find the exact value of the expression.
\\( \tan \left( 30 ^ { \circ } \
ight) = \\)
question help: video ebook message instructor

Explanation:

Step1: Recall the tangent formula

$\tan\theta=\frac{\sin\theta}{\cos\theta}$. For $\theta = 30^{\circ}$, $\sin(30^{\circ})=\frac{1}{2}$ and $\cos(30^{\circ})=\frac{\sqrt{3}}{2}$.

Step2: Calculate the value

$\tan(30^{\circ})=\frac{\sin(30^{\circ})}{\cos(30^{\circ})}=\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}$. When dividing by a fraction, we multiply by its reciprocal, so $\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}=\frac{1}{2}\times\frac{2}{\sqrt{3}}=\frac{1}{\sqrt{3}}$. Rationalizing the denominator (multiplying numerator and denominator by $\sqrt{3}$), we get $\frac{\sqrt{3}}{3}$.

Answer:

$\frac{\sqrt{3}}{3}$