# How do you solve for the unknown lengths and angle measures of triangle ABC where angle C = 90 degrees, angle B = 23 degrees and side a = 24?

May 5, 2018

$A = {90}^{\circ} - B = {67}^{\circ}$

$b = a \tan B \approx 10.19$

$c = \frac{a}{\cos} B \approx 26.07$

#### Explanation:

We have a right triangle, $a = 24 , C = {90}^{\circ} , B = {23}^{\circ} .$

The non-right angles in a right triangle are complementary,

$A = {90}^{\circ} - {23}^{\circ} = {67}^{\circ}$

In a right triangle we have

$\cos B = \frac{a}{c}$

$\tan B = \frac{b}{a}$

so

$b = a \tan B = 24 \tan 23 \approx 10.19$

$c = = \frac{a}{\cos} B = \frac{24}{\cos} 23 \approx 26.07$

May 5, 2018

Refer explanation.

#### Explanation:

Your question indicates unknown lengths which means you want to find length of $b$ and $c$ i assume.

Provided information : Angle B at $23$ degrees // Length of $a$ = $24$ cm

To find length of $c$, use the provided info :

$\sin \left(23\right) = \frac{c}{24}$

$\therefore c = 9.38 c m$ (Rounded off)

When $2$ lengths are found, to find $b$ apply Pythagoras Theorem

$\sqrt{{24}^{2} - {9.38}^{2}}$ = $22.09$ cm ($b$)

To check if our values correspond to the angle given,

${\tan}^{-} 1 \left(\frac{9.28}{22.09}\right) = 23$ degrees sqrt

Since triangle = $180$ degrees, to find angle $A$,

$180 - 23 - 90 = 57$ degrees

May 5, 2018

$\angle A = {67}^{\circ} , b = 10.187 , c = 26.072$

#### Explanation:

$\therefore 180 - \left(90 + 23\right) = {67}^{\circ}$

$\therefore \frac{o p p o s i t e}{a \mathrm{dj} a c e n t} = \tan {23}^{\circ}$

:.opposite=adjacent xx tan 23^

$\therefore o p p o s i t e = 24 \times \tan 23$

$\therefore o p p o s i t e = 10.187 = b$

Pythagoras:-

$\therefore {c}^{2} = {a}^{2} + {b}^{2}$

$\therefore {c}^{2} = {24}^{2} + {10.187}^{2}$

$\therefore {c}^{2} = 576 + 103.775$

$\therefore {c}^{2} = 679.775$

$\therefore \sqrt{{c}^{2}} = \sqrt{679.775}$

$\therefore c = 26.072$