Question

In a triangle is a right angled triangle and $\angle UVW = 90^\circ $, UV = 6 cm, and UW = 8 cm, $\sin W = p:m$, then p + m =?

Answer 1

Hint: Read the question carefully, consider all the given details as it leads to the solution.
Pythagoras theorem can be applied to every right angled triangle.
Pythagoras theorem: gives the relation between the sides of a right angled triangle. It is square of the hypotenuse is equal to the sum of the square of base and square of perpendicular.
 In a right angled triangle, base and perpendicular are at the angle of $90^\circ $ to each other and hypotenuse is the longest side.
In every right angled triangle, relation between the side and its angle is given by trigonometric ratio.
Perpendicular is the side opposite to the angle in trigonometric ratio, hypotenuse is the longest side in a right angled triangle, remaining side is the base.

Complete step by step solution:
Step 1
Draw the figure explained in question

Step 2
Given details:
 in triangle
$\angle UVW = 90^\circ $, UV = 6 cm, and UW = 8 cm,
$\sin W = p:m$

Step 3: find the remaining side of the UVW
UVW is a right angled triangle at V, thus according to
Pythagoras theorem: square of the hypotenuse is equal to the sum of the square of base and square of perpendicular.

Using Pythagoras theorem on UVW
$\mathop {{\rm{Hypotenuse}}}\nolimits^{\rm{2}} {\rm{ = }}\mathop {{\rm{ Base}}}\nolimits^{{\rm{2 }}} {\rm{ + }}\mathop {{\rm{ Perpendicular}}}\nolimits^{\rm{2}} $
      $\mathop {{\rm{UV}}}\nolimits^2 + \mathop {{\rm{VW}}}\nolimits^{\rm{2}} {\rm{ }} = {\rm{ }}\mathop {{\rm{UW}}}\nolimits^{\rm{2}} $
             $\begin{array}{l}
 \Rightarrow \mathop 6\nolimits^2 + \mathop {{\rm{VW}}}\nolimits^{\rm{2}} {\rm{ }} = {\rm{ }}\mathop 8\nolimits^{\rm{2}} \\
 \Rightarrow {\rm{ }}36 + \mathop {{\rm{VW}}}\nolimits^2 = 64\\
 \Rightarrow \mathop {{\rm{VW}}}\nolimits^2 = 64 - 36 = 28
\end{array}$ (given)
                                                                                                             …… (1)

Step 4: we know sin trigonometric relation:
$\sin W = \dfrac{{perpendicular}}{{hypotenuse}}$

${\rm{ }}\sin W = \dfrac{p}{m}$ (given)
Therefore, $\dfrac{{UV}}{{UW}} = \dfrac{p}{m}$
$\begin{array}{l}
 \Rightarrow \dfrac{6}{8} = \dfrac{p}{m}\\
 \Rightarrow \dfrac{3}{4} = \dfrac{p}{m}
\end{array}$
On comparing:
p = 3, m = 4
thus p + m = 3+4
                   = 7

The required answer is p + m = 7.

Note:
Other trigonometric ratios are:
           $\cos \theta = \dfrac{{base}}{{hypotenuse}}$
In this question , \[\cos W = \dfrac{{VW}}{{UW}}\]
                             \[{\rm{ }} = \dfrac{{2\sqrt 7 }}{8} = \dfrac{{\sqrt 7 }}{4}\]
$\tan \theta = \dfrac{{perpendicular}}{{base}}$
In this question, $\tan W = \dfrac{{UV}}{{VW}}$
                                         $\begin{array}{l}
 = \dfrac{6}{{2\sqrt 7 }}\\
 = \dfrac{{3\sqrt 7 }}{7}
\end{array}$