Degrees, minutes, and seconds form is a way to write angle measures using smaller parts of a degree. One degree is split into \(60\) minutes, and one minute is split into \(60\) seconds. These problems practice converting between decimal degrees and D°M’S” form, as well as finding reference angles written with degrees and minutes.
Lesson
Notes
Converting from D°M’S” form to Decimal Form
\(\hspace{25pt}\text{D°M’S”}=D+\displaystyle\frac{M}{60}+\displaystyle\frac{S}{3600}^{\circ}\)
Practice Problems
Convert from D°M’S” form to Decimal Form
\(\textbf{1)}\) \(14^{\circ}\,22{‘}\,17{‘}{‘}\) The answer is approximately \(14.371^{\circ}\)
\(\,\,\,\,\,\,14^{\circ}\,22{‘}\,17{‘}{‘}\)
\(\,\,\,\,\,\,\left(14+\frac{22}{60}+\frac{17}{3600}\right)^{\circ}\)
\(\,\,\,\,\,\,\approx 14.371^{\circ}\)
\(\,\,\,\,\,\,14^{\circ}\,22{‘}\,17{‘}{‘}\)
\(\,\,\,\,\,\,\left(14+\frac{22}{60}+\frac{17}{3600}\right)^{\circ}\)
\(\,\,\,\,\,\,\approx 14.371^{\circ}\)
\(\textbf{2)}\) \(160^{\circ}\,56{‘}\,12{‘}{‘}\) The answer is approximately \(160.937^{\circ}\)
\(\,\,\,\,\,\,160^{\circ}\,56{‘}\,12{‘}{‘}\)
\(\,\,\,\,\,\,\left(160+\frac{56}{60}+\frac{12}{3600}\right)^{\circ}\)
\(\,\,\,\,\,\,\approx 160.937^{\circ}\)
\(\,\,\,\,\,\,160^{\circ}\,56{‘}\,12{‘}{‘}\)
\(\,\,\,\,\,\,\left(160+\frac{56}{60}+\frac{12}{3600}\right)^{\circ}\)
\(\,\,\,\,\,\,\approx 160.937^{\circ}\)
\(\textbf{3)}\) \(12^{\circ}\,35{‘}\,19{‘}{‘}\) The answer is approximately \(12.589^{\circ}\)
\(\,\,\,\,\,\,12^{\circ}\,35{‘}\,19{‘}{‘}\)
\(\,\,\,\,\,\,\left(12+\frac{35}{60}+\frac{19}{3600}\right)^{\circ}\)
\(\,\,\,\,\,\,\approx 12.589^{\circ}\)
\(\,\,\,\,\,\,12^{\circ}\,35{‘}\,19{‘}{‘}\)
\(\,\,\,\,\,\,\left(12+\frac{35}{60}+\frac{19}{3600}\right)^{\circ}\)
\(\,\,\,\,\,\,\approx 12.589^{\circ}\)
\(\textbf{4)}\) \(90^{\circ}\,42{‘}\,37{‘}{‘}\) The answer is approximately \(90.710^{\circ}\)
\(\,\,\,\,\,\,90^{\circ}\,42{‘}\,37{‘}{‘}\)
\(\,\,\,\,\,\,\left(90+\frac{42}{60}+\frac{37}{3600}\right)^{\circ}\)
\(\,\,\,\,\,\,\approx 90.710^{\circ}\)
\(\,\,\,\,\,\,90^{\circ}\,42{‘}\,37{‘}{‘}\)
\(\,\,\,\,\,\,\left(90+\frac{42}{60}+\frac{37}{3600}\right)^{\circ}\)
\(\,\,\,\,\,\,\approx 90.710^{\circ}\)
Convert from decimal form to D°M’S” Form
\(\textbf{5)}\) \(18.45^{\circ}\) The answer is \(18^{\circ}\,27{‘}\,0{‘}{‘}\)
\(\,\,\,\,\,\,18.45^\circ=18^\circ+0.45^\circ\)
\(\,\,\,\,\,\,0.45\cdot 60=27\)
\(\,\,\,\,\,\,\text{So the minutes are }27{‘}.\)
\(\,\,\,\,\,\,0\cdot 60=0\)
\(\,\,\,\,\,\,18.45^\circ=18^\circ\,27{‘}\,0{‘}{‘}\)
\(\,\,\,\,\,\,18.45^\circ=18^\circ+0.45^\circ\)
\(\,\,\,\,\,\,0.45\cdot 60=27\)
\(\,\,\,\,\,\,\text{So the minutes are }27{‘}.\)
\(\,\,\,\,\,\,0\cdot 60=0\)
\(\,\,\,\,\,\,18.45^\circ=18^\circ\,27{‘}\,0{‘}{‘}\)
\(\textbf{6)}\) \(157.82^{\circ}\) The answer is \(157^{\circ}\,49{‘}\,12{‘}{‘}\)
\(\,\,\,\,\,\,157.82^\circ=157^\circ+0.82^\circ\)
\(\,\,\,\,\,\,0.82\cdot 60=49.2\)
\(\,\,\,\,\,\,\text{So the minutes are }49{‘}.\)
\(\,\,\,\,\,\,0.2\cdot 60=12\)
\(\,\,\,\,\,\,157.82^\circ=157^\circ\,49{‘}\,12{‘}{‘}\)
\(\,\,\,\,\,\,157.82^\circ=157^\circ+0.82^\circ\)
\(\,\,\,\,\,\,0.82\cdot 60=49.2\)
\(\,\,\,\,\,\,\text{So the minutes are }49{‘}.\)
\(\,\,\,\,\,\,0.2\cdot 60=12\)
\(\,\,\,\,\,\,157.82^\circ=157^\circ\,49{‘}\,12{‘}{‘}\)
\(\textbf{7)}\) \(130.63^{\circ}\) The answer is \(130^{\circ}\,37{‘}\,48{‘}{‘}\)
\(\,\,\,\,\,\,130.63^\circ=130^\circ+0.63^\circ\)
\(\,\,\,\,\,\,0.63\cdot 60=37.8\)
\(\,\,\,\,\,\,\text{So the minutes are }37{‘}.\)
\(\,\,\,\,\,\,0.8\cdot 60=48\)
\(\,\,\,\,\,\,130.63^\circ=130^\circ\,37{‘}\,48{‘}{‘}\)
\(\,\,\,\,\,\,130.63^\circ=130^\circ+0.63^\circ\)
\(\,\,\,\,\,\,0.63\cdot 60=37.8\)
\(\,\,\,\,\,\,\text{So the minutes are }37{‘}.\)
\(\,\,\,\,\,\,0.8\cdot 60=48\)
\(\,\,\,\,\,\,130.63^\circ=130^\circ\,37{‘}\,48{‘}{‘}\)
\(\textbf{8)}\) \(56.896^{\circ}\) The answer is approximately \(56^{\circ}\,53{‘}\,45.6{‘}{‘}\)
\(\,\,\,\,\,\,56.896^\circ=56^\circ+0.896^\circ\)
\(\,\,\,\,\,\,0.896\cdot 60=53.76\)
\(\,\,\,\,\,\,\text{So the minutes are }53{‘}.\)
\(\,\,\,\,\,\,0.76\cdot 60=45.6\)
\(\,\,\,\,\,\,56.896^\circ\approx56^\circ\,53{‘}\,45.6{‘}{‘}\)
\(\,\,\,\,\,\,56.896^\circ=56^\circ+0.896^\circ\)
\(\,\,\,\,\,\,0.896\cdot 60=53.76\)
\(\,\,\,\,\,\,\text{So the minutes are }53{‘}.\)
\(\,\,\,\,\,\,0.76\cdot 60=45.6\)
\(\,\,\,\,\,\,56.896^\circ\approx56^\circ\,53{‘}\,45.6{‘}{‘}\)
Find the reference angle for the following
\(\textbf{9)}\) \( 154^{\circ} 49′ \) The answer is \( 25^{\circ} 11′\) or \(25.18^{\circ}\)
\(\,\,\,\,\,\,154^\circ49’\text{ is in Quadrant II.}\)
\(\,\,\,\,\,\,\text{Reference angle}=180^\circ-154^\circ49′\)
\(\,\,\,\,\,\,180^\circ00′-154^\circ49’=25^\circ11′\)
\(\,\,\,\,\,\,25^\circ11’=25+\frac{11}{60}\approx25.18^\circ\)
\(\,\,\,\,\,\,154^\circ49’\text{ is in Quadrant II.}\)
\(\,\,\,\,\,\,\text{Reference angle}=180^\circ-154^\circ49′\)
\(\,\,\,\,\,\,180^\circ00′-154^\circ49’=25^\circ11′\)
\(\,\,\,\,\,\,25^\circ11’=25+\frac{11}{60}\approx25.18^\circ\)
\(\textbf{10)}\) Convert \(48^{\circ}\,34{‘}\,21{‘}{‘}\) to decimal form. The answer is approximately \(48.573^\circ\)
\(\,\,\,\,\,\,48^{\circ}\,34{‘}\,21{‘}{‘}\)
\(\,\,\,\,\,\,\left(48+\frac{34}{60}+\frac{21}{3600}\right)^\circ\)
\(\,\,\,\,\,\,48+0.5667+0.0058\approx48.573\)
\(\,\,\,\,\,\,48^{\circ}\,34{‘}\,21{‘}{‘}\approx48.573^\circ\)
\(\,\,\,\,\,\,48^{\circ}\,34{‘}\,21{‘}{‘}\)
\(\,\,\,\,\,\,\left(48+\frac{34}{60}+\frac{21}{3600}\right)^\circ\)
\(\,\,\,\,\,\,48+0.5667+0.0058\approx48.573\)
\(\,\,\,\,\,\,48^{\circ}\,34{‘}\,21{‘}{‘}\approx48.573^\circ\)
\(\textbf{11)}\) Convert \(225^{\circ}\,15{‘}\,30{‘}{‘}\) to decimal form. The answer is approximately \(225.258^\circ\)
\(\,\,\,\,\,\,225^{\circ}\,15{‘}\,30{‘}{‘}\)
\(\,\,\,\,\,\,\left(225+\frac{15}{60}+\frac{30}{3600}\right)^\circ\)
\(\,\,\,\,\,\,225+0.25+0.0083\approx225.258\)
\(\,\,\,\,\,\,225^{\circ}\,15{‘}\,30{‘}{‘}\approx225.258^\circ\)
\(\,\,\,\,\,\,225^{\circ}\,15{‘}\,30{‘}{‘}\)
\(\,\,\,\,\,\,\left(225+\frac{15}{60}+\frac{30}{3600}\right)^\circ\)
\(\,\,\,\,\,\,225+0.25+0.0083\approx225.258\)
\(\,\,\,\,\,\,225^{\circ}\,15{‘}\,30{‘}{‘}\approx225.258^\circ\)
\(\textbf{12)}\) Convert \(42.375^\circ\) to D°M’S” form. The answer is \(42^\circ\,22{‘}\,30{‘}{‘}\)
\(\,\,\,\,\,\,42.375^\circ=42^\circ+0.375^\circ\)
\(\,\,\,\,\,\,0.375\cdot60=22.5\)
\(\,\,\,\,\,\,\text{So the minutes are }22{‘}.\)
\(\,\,\,\,\,\,0.5\cdot60=30\)
\(\,\,\,\,\,\,42.375^\circ=42^\circ\,22{‘}\,30{‘}{‘}\)
\(\,\,\,\,\,\,42.375^\circ=42^\circ+0.375^\circ\)
\(\,\,\,\,\,\,0.375\cdot60=22.5\)
\(\,\,\,\,\,\,\text{So the minutes are }22{‘}.\)
\(\,\,\,\,\,\,0.5\cdot60=30\)
\(\,\,\,\,\,\,42.375^\circ=42^\circ\,22{‘}\,30{‘}{‘}\)
\(\textbf{13)}\) Convert \(73.125^\circ\) to D°M’S” form. The answer is \(73^\circ\,7{‘}\,30{‘}{‘}\)
\(\,\,\,\,\,\,73.125^\circ=73^\circ+0.125^\circ\)
\(\,\,\,\,\,\,0.125\cdot60=7.5\)
\(\,\,\,\,\,\,\text{So the minutes are }7{‘}.\)
\(\,\,\,\,\,\,0.5\cdot60=30\)
\(\,\,\,\,\,\,73.125^\circ=73^\circ\,7{‘}\,30{‘}{‘}\)
\(\,\,\,\,\,\,73.125^\circ=73^\circ+0.125^\circ\)
\(\,\,\,\,\,\,0.125\cdot60=7.5\)
\(\,\,\,\,\,\,\text{So the minutes are }7{‘}.\)
\(\,\,\,\,\,\,0.5\cdot60=30\)
\(\,\,\,\,\,\,73.125^\circ=73^\circ\,7{‘}\,30{‘}{‘}\)
\(\textbf{14)}\) Convert \(112.256^\circ\) to D°M’S” form. The answer is approximately \(112^\circ\,15{‘}\,21.6{‘}{‘}\)
\(\,\,\,\,\,\,112.256^\circ=112^\circ+0.256^\circ\)
\(\,\,\,\,\,\,0.256\cdot60=15.36\)
\(\,\,\,\,\,\,\text{So the minutes are }15{‘}.\)
\(\,\,\,\,\,\,0.36\cdot60=21.6\)
\(\,\,\,\,\,\,112.256^\circ\approx112^\circ\,15{‘}\,21.6{‘}{‘}\)
\(\,\,\,\,\,\,112.256^\circ=112^\circ+0.256^\circ\)
\(\,\,\,\,\,\,0.256\cdot60=15.36\)
\(\,\,\,\,\,\,\text{So the minutes are }15{‘}.\)
\(\,\,\,\,\,\,0.36\cdot60=21.6\)
\(\,\,\,\,\,\,112.256^\circ\approx112^\circ\,15{‘}\,21.6{‘}{‘}\)
\(\textbf{15)}\) Find the reference angle for \(215^\circ30′\). The answer is \(35^\circ30′\)
\(\,\,\,\,\,\,215^\circ30’\text{ is in Quadrant III.}\)
\(\,\,\,\,\,\,\text{Reference angle}=215^\circ30′-180^\circ\)
\(\,\,\,\,\,\,215^\circ30′-180^\circ00’=35^\circ30′\)
\(\,\,\,\,\,\,215^\circ30’\text{ is in Quadrant III.}\)
\(\,\,\,\,\,\,\text{Reference angle}=215^\circ30′-180^\circ\)
\(\,\,\,\,\,\,215^\circ30′-180^\circ00’=35^\circ30′\)
Challenge Problems
\(\textbf{16)}\) Find the reference angle for \(322^\circ18′\). The answer is \(37^\circ42′\)
\(\,\,\,\,\,\,322^\circ18’\text{ is in Quadrant IV.}\)
\(\,\,\,\,\,\,\text{Reference angle}=360^\circ-322^\circ18′\)
\(\,\,\,\,\,\,360^\circ00′-322^\circ18’=37^\circ42′\)
\(\,\,\,\,\,\,322^\circ18’\text{ is in Quadrant IV.}\)
\(\,\,\,\,\,\,\text{Reference angle}=360^\circ-322^\circ18′\)
\(\,\,\,\,\,\,360^\circ00′-322^\circ18’=37^\circ42′\)
\(\textbf{17)}\) Convert \(359^\circ\,59{‘}\,30{‘}{‘}\) to decimal form. The answer is approximately \(359.992^\circ\)
\(\,\,\,\,\,\,359^\circ\,59{‘}\,30{‘}{‘}\)
\(\,\,\,\,\,\,\left(359+\frac{59}{60}+\frac{30}{3600}\right)^\circ\)
\(\,\,\,\,\,\,359+0.9833+0.0083\approx359.992\)
\(\,\,\,\,\,\,359^\circ\,59{‘}\,30{‘}{‘}\approx359.992^\circ\)
\(\,\,\,\,\,\,359^\circ\,59{‘}\,30{‘}{‘}\)
\(\,\,\,\,\,\,\left(359+\frac{59}{60}+\frac{30}{3600}\right)^\circ\)
\(\,\,\,\,\,\,359+0.9833+0.0083\approx359.992\)
\(\,\,\,\,\,\,359^\circ\,59{‘}\,30{‘}{‘}\approx359.992^\circ\)
\(\textbf{18)}\) Convert \(12.999^\circ\) to D°M’S” form. The answer is approximately \(12^\circ\,59{‘}\,56.4{‘}{‘}\)
\(\,\,\,\,\,\,12.999^\circ=12^\circ+0.999^\circ\)
\(\,\,\,\,\,\,0.999\cdot60=59.94\)
\(\,\,\,\,\,\,\text{So the minutes are }59{‘}.\)
\(\,\,\,\,\,\,0.94\cdot60=56.4\)
\(\,\,\,\,\,\,12.999^\circ\approx12^\circ\,59{‘}\,56.4{‘}{‘}\)
\(\,\,\,\,\,\,12.999^\circ=12^\circ+0.999^\circ\)
\(\,\,\,\,\,\,0.999\cdot60=59.94\)
\(\,\,\,\,\,\,\text{So the minutes are }59{‘}.\)
\(\,\,\,\,\,\,0.94\cdot60=56.4\)
\(\,\,\,\,\,\,12.999^\circ\approx12^\circ\,59{‘}\,56.4{‘}{‘}\)
\(\textbf{19)}\) Find the reference angle for \(278^\circ42′\). The answer is \(81^\circ18′\)
\(\,\,\,\,\,\,278^\circ42’\text{ is in Quadrant IV.}\)
\(\,\,\,\,\,\,\text{Reference angle}=360^\circ-278^\circ42′\)
\(\,\,\,\,\,\,360^\circ00′-278^\circ42’=81^\circ18′\)
\(\,\,\,\,\,\,278^\circ42’\text{ is in Quadrant IV.}\)
\(\,\,\,\,\,\,\text{Reference angle}=360^\circ-278^\circ42′\)
\(\,\,\,\,\,\,360^\circ00′-278^\circ42’=81^\circ18′\)
\(\textbf{20)}\) Convert \(24^\circ\,37{‘}\,12{‘}{‘}\) to decimal form. The answer is \(24.62^\circ\)
\(\,\,\,\,\,\,24^\circ\,37{‘}\,12{‘}{‘}\)
\(\,\,\,\,\,\,\left(24+\frac{37}{60}+\frac{12}{3600}\right)^\circ\)
\(\,\,\,\,\,\,24+0.6167+0.0033=24.62\)
\(\,\,\,\,\,\,24^\circ\,37{‘}\,12{‘}{‘}=24.62^\circ\)
\(\,\,\,\,\,\,24^\circ\,37{‘}\,12{‘}{‘}\)
\(\,\,\,\,\,\,\left(24+\frac{37}{60}+\frac{12}{3600}\right)^\circ\)
\(\,\,\,\,\,\,24+0.6167+0.0033=24.62\)
\(\,\,\,\,\,\,24^\circ\,37{‘}\,12{‘}{‘}=24.62^\circ\)
See Related Pages\(\)
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\(\bullet\text{ Angle of Depression and Elevation}\)
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\(\bullet\text{ Convert to Radians and to Degrees}\)
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\(\bullet\text{ Degrees, Minutes and Seconds}\)
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\(\bullet\text{ Coterminal Angles}\)
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\(\bullet\text{ Reference Angles}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
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\(…\)
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\(\,\,\,\,\,\,\,\,\sin{(60^{\circ})}=\displaystyle\frac{\sqrt{3}}{2}…\)
\(\bullet\text{ Law of Sines}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{\sin{A}}{a}=\frac{\sin{B}}{b}=\frac{\sin{C}}{c}\) 
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\(\bullet\text{ Area of SAS Triangles}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\frac{1}{2}ab \sin{C}\) \(…\)
\(\bullet\text{ Law of Cosines}\)
\(\,\,\,\,\,\,\,\,a^2=b^2+c^2-2bc \cos{A}\) \(…\)
\(\bullet\text{ Area of SSS Triangles (Heron’s formula)}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\sqrt{s(s-a)(s-b)(s-c)}\) \(…\)
\(\bullet\text{ Geometric Mean}\)
\(\,\,\,\,\,\,\,\,x=\sqrt{ab} \text{ or } \displaystyle\frac{a}{x}=\frac{x}{b}…\)
\(\bullet\text{ Geometric Mean- Similar Right Triangles}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
\(\bullet\text{ Inverse Trigonmetric Functions}\)
\(\,\,\,\,\,\,\,\,\sin {\left(cos^{-1}\left(\frac{3}{5}\right)\right)}…\)
\(\bullet\text{ Sum and Difference of Angles Formulas}\)
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\(\bullet\text{ Double-Angle and Half-Angle Formulas}\)
\(\,\,\,\,\,\,\,\,\sin{(2A)}=2\sin{(A)}\cos{(A)}…\)
\(\bullet\text{ Trigonometry-Pythagorean Identities}\)
\(\,\,\,\,\,\,\,\,\sin^2{(x)}+\cos^2{(x)}=1…\)
\(\bullet\text{ Product-Sum Identities}\)
\(\,\,\,\,\,\,\,\,\cos{\alpha}\cos{\beta}=\left(\displaystyle\frac{\cos{(\alpha+\beta)}+\cos{(\alpha-\beta)}}{2}\right)…\)
\(\bullet\text{ Cofunction Identities}\)
\(\,\,\,\,\,\,\,\,\sin{(x)}=\cos{(\frac{\pi}{2}-x)}…\)
\(\bullet\text{ Proving Trigonometric Identities}\)
\(\,\,\,\,\,\,\,\,\sec{x}-\cos{x}=\displaystyle\frac{\tan^2{x}}{\sec{x}}…\)
\(\bullet\text{ Graphing Trig Functions- sin and cos}\)
\(\,\,\,\,\,\,\,\,f(x)=A \sin{B(x-c)}+D \) 
\(…\)
\(\bullet\text{ Solving Trigonometric Equations}\)
\(\,\,\,\,\,\,\,\,2\cos{(x)}=\sqrt{3}…\)