Question

consider right triangle $\triangle mno$ below.

which expressions represent the length of side $\overline{mn}$?
choose 2 answers:
a $4 \cdot \sin(65^{\circ})$
b $1.9 \cdot \cos(65^{\circ})$
c $\frac{4}{\sin(65^{\circ})}$
d $\frac{1.9}{\cos(65^{\circ})}$

Explanation:

Step1: Calculate ∠M

First, find angle at M:
$\angle M = 90^\circ - 25^\circ = 65^\circ$

Step2: Use sine for ∠N

Relate MN, NO, ∠N:
$\sin(25^\circ) = \frac{MO}{MN} \implies MN = \frac{MO}{\sin(25^\circ)}$
Since $\sin(25^\circ)=\cos(65^\circ)$, substitute:
$MN = \frac{1.9}{\cos(65^\circ)}$

Step3: Use sine for ∠M

Relate MN, NO, ∠M:
$\sin(65^\circ) = \frac{NO}{MN} \implies MN = \frac{NO}{\sin(65^\circ)} = \frac{4}{\sin(65^\circ)}$

Answer:

C. $\frac{4}{\sin(65^\circ)}$
D. $\frac{1.9}{\cos(65^\circ)}$