Question

In ∆NOP, o =3.6 cm,

Answer 1

First, notice that since the sum of the internal angles of any triangle must always be equal to 180, then:

`\begin{gathered} \angle N+\angle O+\angle P=180 \\ \Rightarrow119+5+\angle P=180 \\ \Rightarrow\angle P=180-119-5 \\ \Rightarrow\angle P=56 \end{gathered}`

Draw a diagram of the triangle to visualize the situation:

Draw the altitude of the triangle through the vertex N. Let H be the point where the altitude through N intercepts the line OP. From the trigonometric relations, notice the following:

`\begin{gathered} \tan (O)=(HN)/(OH) \\ \cos (P)=(HP)/(NP) \\ \sin (P)=(HN)/(NP) \end{gathered}`

Substitute the known values:

Isolate OH from the first equation, HP from the second and h from the third one:

Substitute the expression for h from the third equation into the expression for OH:

Finally, notice that the length of n equals the length of the segment OP, which is the sum of the lengths of the segments OH and HP:

Therefore, to the nearest tenth: