Question

The circular measure of two angles of a triangle are $\dfrac{1}{2}$ and $\dfrac{1}{3}$ respectively, what is the number of degrees in the third angle? 

Answer 1

Hint: By the help of the theorem stating that the sum of the interior angle of a triangle is equal to ${{180}^{\circ }}$ we will find the third angle. Also, we will use the formula which is given by ${{\left( \pi  \right)}^{c}}={{180}^{\circ }}$ to solve the question. This can also be written as ${{\left( 1 \right)}^{c}}={{\left( \dfrac{180}{\pi } \right)}^{\circ }}$ after dividing the equation by $\pi $.

Complete step-by-step answer:

In the question we are given the two angles of a triangle which are $\dfrac{1}{2}$ and $\dfrac{1}{3}$. As we are not directly given in what form we have $\dfrac{1}{2}$ and $\dfrac{1}{3}$. Therefore, we will consider them as radians. Thus we have ${{\left( \dfrac{1}{2} \right)}^{c}}$ and ${{\left( \dfrac{1}{3} \right)}^{c}}$. Now, we are given the two angles out of three therefore, we will consider the third angle as x degree. As we know that the sum of the interior angle of a triangle is equal to ${{180}^{\circ }}$. Therefore, we have

${{\left( \dfrac{1}{2} \right)}^{c}}+{{\left( \dfrac{1}{3} \right)}^{c}}+{{\left( x \right)}^{\circ }}={{180}^{\circ }}...(i)$

Now, we will first convert ${{\left( \dfrac{1}{2} \right)}^{c}}$ into degrees. This can be done by the formula given by ${{\left( \pi  \right)}^{c}}={{180}^{\circ }}$ or, ${{\left( 1 \right)}^{c}}={{\left( \dfrac{180}{\pi } \right)}^{\circ }}$. We can write ${{\left( \dfrac{1}{2} \right)}^{c}}$ as ${{\left( \dfrac{1}{2} \right)}^{c}}=\dfrac{1}{2}\times {{\left( 1 \right)}^{c}}$. By substituting ${{\left( 1 \right)}^{c}}={{\left( \dfrac{180}{\pi } \right)}^{\circ }}$ we will get

$ {{\left( \dfrac{1}{2} \right)}^{c}}=\dfrac{1}{2}\times {{\left( 1 \right)}^{c}} $

$ \Rightarrow {{\left( \dfrac{1}{2} \right)}^{c}}=\dfrac{1}{2}\times {{\left( \dfrac{180}{\pi } \right)}^{\circ }} $

$ \Rightarrow {{\left( \dfrac{1}{2} \right)}^{c}}={{\left( \dfrac{1}{2}\times \dfrac{180}{\pi } \right)}^{\circ }} $

$ \Rightarrow {{\left( \dfrac{1}{2} \right)}^{c}}={{\left( \dfrac{1}{1}\times \dfrac{90}{\pi } \right)}^{\circ }} $

$ \Rightarrow {{\left( \dfrac{1}{2} \right)}^{c}}={{\left( \dfrac{90}{\pi } \right)}^{\circ }} $

And now we will first convert ${{\left( \dfrac{1}{3} \right)}^{c}}$ into degrees. This can be done by the formula given by ${{\left( \pi  \right)}^{c}}={{180}^{\circ }}$ or, ${{\left( 1 \right)}^{c}}={{\left( \dfrac{180}{\pi } \right)}^{\circ }}$. We can write ${{\left( \dfrac{1}{3} \right)}^{c}}$ as ${{\left( \dfrac{1}{3} \right)}^{c}}=\dfrac{1}{3}\times {{\left( 1 \right)}^{c}}$. By substituting ${{\left( 1 \right)}^{c}}={{\left( \dfrac{180}{\pi } \right)}^{\circ }}$ we will get

$ {{\left( \dfrac{1}{3} \right)}^{c}}=\dfrac{1}{3}\times {{\left( 1 \right)}^{c}} $

$ \Rightarrow {{\left( \dfrac{1}{3} \right)}^{c}}=\dfrac{1}{3}\times {{\left( \dfrac{180}{\pi } \right)}^{\circ }} $

$ \Rightarrow {{\left( \dfrac{1}{3} \right)}^{c}}={{\left( \dfrac{1}{3}\times \dfrac{180}{\pi } \right)}^{\circ }} $ 

$ \Rightarrow {{\left( \dfrac{1}{3} \right)}^{c}}={{\left( \dfrac{1}{1}\times \dfrac{60}{\pi } \right)}^{\circ }} $

$ \Rightarrow {{\left( \dfrac{1}{3} \right)}^{c}}={{\left( \dfrac{60}{\pi } \right)}^{\circ }} $

Now, we will substitute the value of ${{\left( \dfrac{1}{2} \right)}^{c}}={{\left( \dfrac{90}{\pi } \right)}^{\circ }}$ and ${{\left( \dfrac{1}{3} \right)}^{c}}={{\left( \dfrac{60}{\pi } \right)}^{\circ }}$ in equation (i). Thus, we have

$ {{\left( \dfrac{1}{2} \right)}^{c}}+{{\left( \dfrac{1}{3} \right)}^{c}}+{{\left( x \right)}^{\circ }}={{180}^{\circ }} $

$ \Rightarrow {{\left( \dfrac{90}{\pi } \right)}^{\circ }}+{{\left( \dfrac{60}{\pi } \right)}^{\circ }}+{{\left( x \right)}^{\circ }}={{180}^{\circ }} $

$ \Rightarrow {{\left( \dfrac{90}{\pi }+\dfrac{60}{\pi }+x \right)}^{\circ }}={{180}^{\circ }} $

$ \Rightarrow {{\left( \dfrac{90+60+x\pi }{\pi } \right)}^{\circ }}={{180}^{\circ }} $

$ \Rightarrow \dfrac{90+60+x\pi }{\pi }=180 $

$ \Rightarrow 150+x\pi =180\pi  $

$ \Rightarrow x\pi =180\pi -150 $ 

$ \Rightarrow x=\dfrac{180\pi }{\pi }-\dfrac{150}{\pi } $

$ \Rightarrow x=180-\dfrac{150}{\pi } $

Now, we will put the value of $\pi =3.142$ in order to solve the problem further. So, now we get

$x=180-\dfrac{150}{\pi } $

$ \Rightarrow x=180-\dfrac{150}{3.142} $

$ \Rightarrow x=180-47.74  $

$ \Rightarrow x=132.26 $

Hence, the third angle is $x={{132.26}^{\circ }}$. 

Note: While substituting $\pi ={{180}^{\circ }}$ we will mind that we are not placing this value in the formula but to solve the problem further. Always try to write formulas in a number or a decimal rather than a fraction. Sometimes we are not given whether we are given radians or degrees. So, we will  consider them as radians only.