Question

find tanθ, where θ is the angle shown. give an exact value, not a decimal approximation.

Explanation:

Step1: Find the adjacent side

Use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 13\) and \(b = 12\). Let the adjacent side to \(\theta\) be \(a\). Then \(a=\sqrt{c^{2}-b^{2}}=\sqrt{13^{2}-12^{2}}=\sqrt{169 - 144}=\sqrt{25}=5\).

Step2: Recall the tangent - ratio formula

The formula for \(\tan\theta\) in a right - triangle is \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). The opposite side to \(\theta\) is 12 and the adjacent side is 5.

Step3: Calculate \(\tan\theta\)

\(\tan\theta=\frac{12}{5}\)

Answer:

\(\frac{12}{5}\)