Question

Find the missing side lengths. Leave your answers as radicals in simplest form.

Answer 1

Answer:

D.

$u =3$

$v =\sqrt{3}$

Step-by-step explanation:

Given

The triangle in the diagram above

Required

Find the missing lengths, u and v

To find the missing lengths, we have to check the relationship between the missing lengths, the given length and the given angle;

From trigonometry;

$sin\ \theta = \frac{Opp}{Hyp}$; Where Opp = Opposite and Hyp = Hypotenuse

From the attached diagram;

$\theta = 30$

$Opp = v$

$Hyp = 2\sqrt{3}$

$sin\ \theta = \frac{Opp}{Hyp}$  becomes

$sin\ 30 = \frac{v}{2\sqrt{3}}$

Multiply both sides by $2\sqrt{3}$

$2\sqrt{3} * sin\ 30 = \frac{v}{2\sqrt{3}} * 2\sqrt{3}$

$2\sqrt{3} * sin\ 30 = v$

In radians, $sin30 = \frac{1}{2}$

$2\sqrt{3} * \frac{1}{2} = v$

$\sqrt{3} = v$

$v =\sqrt{3}$

Similarly, From trigonometry;

$cos\ \theta = \frac{Adj}{Hyp}$; Where Adj = Adjacent

From the attached diagram;

$\theta = 30$

$Adj = u$

$Hyp = 2\sqrt{3}$

$cos\ \theta = \frac{Adj}{Hyp}$  becomes

$cos\ 30 = \frac{u}{2\sqrt{3}}$

Multiply both sides by $2\sqrt{3}$

$2\sqrt{3} * cos\ 30 = \frac{u}{2\sqrt{3}} * 2\sqrt{3}$

$2\sqrt{3} * cos\ 30 = u$

In radians, $cos30 = \frac{\sqrt{3}}{2}$

$2\sqrt{3} * \frac{\sqrt{3}}{2} = u$

$\sqrt{3} * \sqrt{3} = u$

$3 = u$

$u =3$