Question

A young man makes a corner bookshelf for his textbooks in an inside corner of his living room. The lengths of the sides of the corner bookshelf are 18 inches, 18inches, and 16 inches. What are the respective angles, in degrees? Round to the nearest hundredth.ne

Answer 1

$\begin{gathered} \\ 63.61\text{ \degree ,63.61 \degree and 52.78 \degree} \end{gathered}$

Explanation

to solve this we can use the law of cosines,it says

$\begin{gathered} a^2=b^2+c^2-2bc\text{ *cos\lparen A\rparen} \\ b^2=a^2+c^2-2ac*\text{cos}\operatorname{\lparen}B\operatorname{\rparen} \\ c^2=a^2+b^2-2ab*\text{cos}\operatorname{\lparen}C\operatorname{\rparen} \end{gathered}$

Step 1

hence, let

$\begin{gathered} a=\text{ 18} \\ b=18 \\ c=16 \end{gathered}$

now, let's find the angles

a) angle A)

$\begin{gathered} a^{2}=b^{2}+c^{2}-2bc\text{cos\operatorname{\lparen}A\operatorname{\rparen}} \\ 18^2=18^2+16^2-2\left(18\right)\left(16\right)cos\lparen A) \\ 0=16^2-576*Cos\left(A\right) \\ -256=-576*Cos\left(A\right) \\ divide\text{ both sides by -576} \\ \frac{-256}{-576}=\frac{-576Cos(A)}{-576} \\ 0.444=cos\left(A\right? \\ \cos^{-1}\mleft(0.444\mright)=A \\ A=63.61\text{ \degree} \end{gathered}$

so

the first angle is 63.61 °

b) angle B)

$\begin{gathered} b^2=a^2+c^2-2ac\text{cos}\operatorname{\lparen}B\operatorname{\rparen} \\ 18^2=18^2+16^2-2\left(18\right)\left(16\right)cos\lparen B) \\ 0=16^2-576*Cos\left(B\right) \\ -256=-576*Cos\left(B\right) \\ divide\text{ both sides by -576} \\ \frac{-256}{-576}=\frac{-576Cos(B)}{-576} \\ 0.444=cos\left(B\right? \\ \cos^{-1}\mleft(0.444\mright)=B \\ B=63.61\text{ \degree} \end{gathered}$

so,the second angle is 63.61

c)angle C)

$\begin{gathered} c^2=a^2+b^2-2ab\text{cos}\operatorname{\lparen}C\operatorname{\rparen} \\ 16^2=18^2+18^2-2\left(18\right)\left(18\right)cos\lparen C) \\ 256=648-648cos\lparen C) \\ 256-648=-648*Cos\left(C\right) \\ divide\text{ both sides by -648} \\ \frac{-392}{-648}=\frac{-648Cos(C)}{-648} \\ 0.604=cos\left(C\right? \\ \cos^{-1}\mleft(0.604\mright)=C \\ C=52.78\text{ \degree} \end{gathered}$

so, the entire answer is

$\begin{gathered} \\ 63.61\text{ \degree ,63.61 \degree and 52.78 \degree} \end{gathered}$

I hope this helps you