Answer

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**Hint:**a. To convert from degree to radian in this question, we will start to convert all other given data into degree by using proper mathematical operations and then we will convert obtained number in degree from degree to radian measure by multiplying $\dfrac{\pi }{180^\circ }$.

b. To convert from degree to radian in the given question, we convert all the other data into degree with help of appropriate mathematical operations and then we will convert obtained value in degree into centesimal system by multiplying ${{\left( \dfrac{100}{90^\circ } \right)}^{g}}$.

**Complete step by step answer:**

a. Since, the given that the number in degree:

$\Rightarrow 55^\circ 16'30''$

Now, we will convert $16'$ and $30''$ into degree by dividing $60$ and $3600$ respectively as:

$\Rightarrow 55^\circ +\left( \dfrac{16}{60} \right)^\circ +\left( \dfrac{30}{3600} \right)^\circ $

Here, we will find the simplest form of fraction as:

$\Rightarrow 55^\circ +\left( \dfrac{4}{15} \right)^\circ +\left( \dfrac{1}{120} \right)^\circ $

Now, we will multiply by $\dfrac{\pi }{180^\circ }$ in the above step to convert it into radian measure as:

\[\Rightarrow 55^\circ \times \dfrac{\pi }{180^\circ }+\left( \dfrac{4}{15} \right)^\circ \times \dfrac{\pi }{180^\circ }+\left( \dfrac{1}{120} \right)^\circ \times \dfrac{\pi }{180^\circ }\]

Simplify the above as:

\[\Rightarrow 11\times \dfrac{\pi }{36}+\left( \dfrac{1}{15} \right)\times \dfrac{\pi }{45}+\left( \dfrac{1}{120} \right)\times \dfrac{\pi }{180}\]

Here, we will do the multiplication of fractions as:

\[\Rightarrow \dfrac{11\pi }{36}+\dfrac{\pi }{675}+\dfrac{\pi }{21600}\]

Now, we will use the addition of fraction. So, we will take L.C.M. of all denominators and it will be $21600$. So, we can write above step as:

\[\Rightarrow \dfrac{600\times 11\pi +32\times \pi +\pi }{21600}\]

Here, we will do multiplication as:

\[\Rightarrow \dfrac{6600\pi +32\pi +\pi }{21600}\]

Now, we will do addition in numerator as:

\[\Rightarrow \dfrac{6633\pi }{21600}\]

Here, we will convert the fraction into decimal number up to two places as:

$\Rightarrow 0.31\pi $

As we know that the value of $\pi $ is $3.14$. We will substitute $3.14$ for $\pi $ in the above step as:

$\Rightarrow 0.31\times 3.14$

After multiplying $0.31$ and $3.14$, we will have $0.9734$ up to four decimal places as:

$\Rightarrow 0.9734$radian.

Hence, Conversion of $55^\circ 16'30''$ into radian measure is $0.9734$radian.

b. Since, the given that the number in degree:

$\Rightarrow 10^\circ 10'10''$

Now, we will convert $10'$ and $10''$ into degree by dividing $60$ and $3600$ respectively as:

$\Rightarrow 10^\circ +\left( \dfrac{10}{60} \right)^\circ +\left( \dfrac{10}{3600} \right)^\circ $

Here, we will find the simplest form of fraction as:

$\Rightarrow 10^\circ +\left( \dfrac{1}{6} \right)^\circ +\left( \dfrac{1}{3600} \right)^\circ $

Now, we will multiply by ${{\left( \dfrac{100}{90^\circ } \right)}^{g}}$ in the above step to convert it into centesimal measure as:

$\Rightarrow 10^\circ \times {{\left( \dfrac{100}{90^\circ } \right)}^{g}}+\left( \dfrac{1}{6} \right)^\circ \times {{\left( \dfrac{100}{90^\circ } \right)}^{g}}+\left( \dfrac{1}{3600} \right)^\circ \times {{\left( \dfrac{100}{90^\circ } \right)}^{g}}$

Simplify the above as:

$\begin{align}

& \Rightarrow {{\left( 10^\circ \times \dfrac{100}{90^\circ } \right)}^{g}}+{{\left( \left( \dfrac{1}{6} \right)^\circ \times \dfrac{100}{90^\circ } \right)}^{g}}+{{\left( \left( \dfrac{1}{3600} \right)^\circ \times \dfrac{100}{90^\circ } \right)}^{g}} \\

& \Rightarrow {{\left( \dfrac{100}{9} \right)}^{g}}+{{\left( \dfrac{100}{540} \right)}^{g}}+{{\left( \dfrac{100}{324000} \right)}^{g}} \\

\end{align}$

We can write the above step as:

$\Rightarrow {{\left( \dfrac{100}{9}+\dfrac{100}{540}+\dfrac{100}{324000} \right)}^{g}}$

Here, we will convert all the fractions into decimal numbers as:

$\Rightarrow {{\left( 11.1111111+0.1851851+0.0003086 \right)}^{g}}$

After addition of the numbers of above step, we will have:

$\Rightarrow {{\left( 11.2966048 \right)}^{g}}$

After multiplying $0.31$ and $3.14$, we can write the above decimal number up to four decimal places as:

$\Rightarrow {{\left( 11.2966 \right)}^{g}}$

Hence, Conversion of $10^\circ 10'10''$ into centesimal measure is $11.2966$grade.

**Note:**a. Here, we will understand why we multiply by $\dfrac{\pi }{180^\circ }$ for radian measure:

Since, $180^\circ $ is equal to $\pi $ as:

$\Rightarrow 180^\circ =\pi $

Now, we will divide by $180^\circ $ both sides as:

\[\Rightarrow \dfrac{180^\circ }{180^\circ }=\dfrac{\pi }{180^\circ }\]

After solving it, we will have:

\[\Rightarrow 1^\circ =\dfrac{\pi }{180^\circ }\]

Now, we will convert $x$ into radian as:

\[\Rightarrow x\times 1^\circ =x\times \dfrac{\pi }{180^\circ }\]

\[\Rightarrow x^\circ =x\times \dfrac{\pi }{180^\circ }\]

This is the formula used for converting any degree value to radian measure.

b. Here, we will understand why we multiply by $\dfrac{100}{90^\circ }$ for centesimal measure:

Since, $90^\circ $ is equal to $100$ grade as:

$\Rightarrow 90^\circ ={{100}^{g}}$

Now, we will divide by $90^\circ $ both sides as:

\[\Rightarrow \dfrac{90^\circ }{90^\circ }=\dfrac{{{100}^{g}}}{90^\circ }\]

After solving it, we will have:

\[\Rightarrow 1^\circ =\dfrac{{{100}^{g}}}{90^\circ }\]

Now, we can convert $x$ into centesimal as:

\[\Rightarrow x\times 1^\circ =x\times \dfrac{{{100}^{g}}}{90^\circ }\]

\[\Rightarrow x^\circ =x\times \dfrac{{{100}^{g}}}{90^\circ }\]

This is the formula used for converting any degree value to centesimal measure.

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