Question

find the exact value of tan k in simplest radical form.
(diagram: right triangle mlk with right angle at l, ml = 7, mk = 10, lk = √51)

Explanation:

Step1: Recall the definition of tangent in a right triangle

In a right triangle, the tangent of an acute angle is the ratio of the length of the opposite side to the length of the adjacent side. For angle \( K \) in right triangle \( KLM \) (right - angled at \( L \)):
\(\tan K=\frac{\text{opposite side to }K}{\text{adjacent side to }K}\)
The side opposite to angle \( K \) is \( ML \) with length \( 7 \), and the side adjacent to angle \( K \) is \( LK \) with length \( \sqrt{51} \).

Step2: Calculate \(\tan K\)

Substitute the lengths of the opposite and adjacent sides into the tangent formula:
\(\tan K = \frac{ML}{LK}=\frac{7}{\sqrt{51}}\)
To rationalize the denominator, we multiply the numerator and the denominator by \(\sqrt{51}\):
\(\tan K=\frac{7\times\sqrt{51}}{\sqrt{51}\times\sqrt{51}}=\frac{7\sqrt{51}}{51}\)

Answer:

\(\frac{7\sqrt{51}}{51}\)