Question

find the exact value of the following expression.

(\tan left(\frac{pi}{6}+\frac{5 pi}{4}
ight))

find the exact value of the expression.

(\tan left(\frac{pi}{6}+\frac{5 pi}{4}
ight)=)
(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression. rationalize all denominators.)

Explanation:

Step1: Use the tangent addition formula

The tangent addition formula is $\tan(A + B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$. Here $A = \frac{\pi}{6}$ and $B=\frac{5\pi}{4}$.
We know that $\tan\frac{\pi}{6}=\frac{\sqrt{3}}{3}$ and $\tan\frac{5\pi}{4}=\tan(\pi+\frac{\pi}{4})=\tan\frac{\pi}{4} = 1$.
So, $\tan(\frac{\pi}{6}+\frac{5\pi}{4})=\frac{\tan\frac{\pi}{6}+\tan\frac{5\pi}{4}}{1 - \tan\frac{\pi}{6}\tan\frac{5\pi}{4}}$.

Step2: Substitute the values

Substitute $\tan\frac{\pi}{6}=\frac{\sqrt{3}}{3}$ and $\tan\frac{5\pi}{4} = 1$ into the formula:

$$
LATEXBLOCK0
$$

Step3: Rationalize the denominator

Multiply the numerator and denominator by the conjugate of the denominator $3 + \sqrt{3}$:

$$
LATEXBLOCK1
$$

Answer:

$\sqrt{3}+2$