Q. 48 - Precalculus Enhanced with Graphing Utilities

Question

(a)Convert the angle$32°{20}^{\text{'}}{35}^{\text{'}\text{'}}$ to a decimal in degrees. Round the answer to two decimal places.

(b)Convert the angle $63.18°$ to $\mathrm{D}°{\mathrm{M}}^{\text{'}}{\mathrm{S}}^{\text{'}\text{'}}$ form. Express the answer to the nearest second.

Step-by-Step Solution

Answer

(a) The angle in degrees is $32.3397$

(b) The angle in DMS is $63°{10}^{\text{'}}{48}^{\text{'}\text{'}}$

Part (a) Step 1: Given information

Given the angle $32°{20}^{\text{'}}{35}^{\text{'}\text{'}}$

Part (b) Step 2: Use conversion equivalents and calculate

Here

$$\begin{gathered} {\mathrm{I}}^{\text{'}}=\left(\frac{1}{60}\right) \\ {1}^{\text{'}\text{'}}={\left(\frac{1}{60}\right)}^{\text{'}} \\ =\left(\frac{1}{60}·\frac{1}{60}\right) \end{gathered}$$

$$\begin{gathered} 32°{20}^{\text{'}}{35}^{\text{'}\text{'}}=32°+{20}^{\text{'}}\left(\frac{1}{60}\right)+{35}^{\text{'}\text{'}}\left(\frac{1}{60}·\frac{1}{60}\right) \\ 32°{20}^{\text{'}}{35}^{\text{'}\text{'}}=32°+0.33°+0.0097° \\ 32°{20}^{\text{'}}{35}^{\text{'}\text{'}}\approx 32.3397 \end{gathered}$$

Part (b) Step 1: Given information

Given the angle $63.18°$

Part (b) Step 2: Use conversion equivalents and calculate

Calculating, we get

$$\begin{gathered} 1°={60}^{\text{'}} \\ {l}^{\text{'}}={60}^{\text{'}} \end{gathered}$$

$$\begin{gathered} 63.18°=63°+(0.18)·1° \\ 63.18°=63°+(0.18)·{60}^{\text{'}} \\ 63.18°=63°+10.8^{\text{'}} \\ 63.18°=63°+{10}^{\text{'}}+(0.8)·{60}^{\text{'}\text{'}} \\ 63.18°=63°+{10}^{\text{'}}+{48}^{\text{'}\text{'}} \\ 63.18°=63°{10}^{\text{'}}{48}^{\text{'}\text{'}} \end{gathered}$$