Question

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

Answer 1

Step-1: To evaluate the third angle of the triangle:

The angle $C$ is given as $90$ degrees , angle $B$ is given as $23$ degrees and side $a$ is given as $24$.

The sum of the interior angles of a triangle is $180\mathrm{degrees}$.

Hence,

The sum of the angles of the triangle $A,B$ and $C$ is $180\mathrm{degrees}$ so the value of the third angle $C$ will be:

$$\begin{gathered} A+B+C=180° \\ \Rightarrow A+23°+90°=180° \\ \Rightarrow A+113°=180° \\ \Rightarrow A=180°-113° \\ \Rightarrow A=67° \end{gathered}$$

Hence the third angle is of $67°$.

Step-2: To evaluate the missing sides of the triangle:

The formula to find the side $b$ of the triangle is $b=a\tan B$,

$$\begin{gathered} b=24\tan 23° \\ \simeq 10.19 \end{gathered}$$

The formula to find the side $c$ of the triangle is $c=\frac{a}{\cos B}$,

$$\begin{gathered} c=\frac{a}{\cos B} \\ =\frac{24}{\cos 23} \\ \simeq 26.07 \end{gathered}$$

Hence, the length of the missing sides are $10.19$ and $26.07$.