Question

trigonometric ratios
deriving a tangent ratio for special triangles
tan(45°) = \frac{sqrt{2}}{2}
nm = x
tan(45°) = 1

Explanation:

Step1: Recall tangent - ratio formula

The tangent of an angle in a right - triangle is defined as $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. In right - triangle $LMN$ with $\angle L = 45^{\circ}$ and $\angle M=45^{\circ}$, if one of the legs (say $LN = x$), since the two non - right angles are equal, the other leg $NM=x$ (because in a $45 - 45-90$ triangle, the two legs are of equal length).

Step2: Calculate $\tan(45^{\circ})$

Using the formula $\tan(45^{\circ})=\frac{LN}{NM}$, and since $LN = x$ and $NM = x$, then $\tan(45^{\circ})=\frac{x}{x}=1$.

Answer:

$\tan(45^{\circ}) = 1$