Converting between degrees and radians is an important skill for working with angles in trigonometry and the unit circle. To convert degrees to radians, multiply by \(\frac{\pi}{180^\circ}\), and to convert radians to degrees, multiply by \(\frac{180^\circ}{\pi}\). These problems practice both directions, including positive angles, negative angles, improper radian measures, and common unit-circle angles.
Notes
Practice Problems
\(\textbf{1)}\) Convert \( \theta=55^{\circ} \) from degrees to radiansThe answer is \( \displaystyle\frac{11\pi}{36} \) radians
\(\text{Degrees } \rightarrow \text{ Radians}\)
\(\,\,\,\,\, 55^{\circ} \times \displaystyle \frac{\pi}{180^{\circ}}=\frac{55\pi}{180}=\frac{11\pi}{36}\)
\(\text{Degrees } \rightarrow \text{ Radians}\)
\(\,\,\,\,\, 55^{\circ} \times \displaystyle \frac{\pi}{180^{\circ}}=\frac{55\pi}{180}=\frac{11\pi}{36}\)
\(\textbf{2)}\) Convert \( \theta=270^{\circ} \) from degrees to radians The answer is \( \displaystyle\frac{3\pi}{2} \) radians
\(\text{Degrees } \rightarrow \text{ Radians}\)
\(\,\,\,\,\, 270^{\circ} \times \displaystyle \frac{\pi}{180^{\circ}}=\frac{270\pi}{180}=\frac{3\pi}{2}\)
\(\text{Degrees } \rightarrow \text{ Radians}\)
\(\,\,\,\,\, 270^{\circ} \times \displaystyle \frac{\pi}{180^{\circ}}=\frac{270\pi}{180}=\frac{3\pi}{2}\)
\(\textbf{3)}\) Convert \( \theta=-275^{\circ} \) from degrees to radians The answer is \( – \displaystyle\frac{55\pi}{36} \) radians
\(\text{Degrees } \rightarrow \text{ Radians}\)
\(\,\,\,\,\, -275^{\circ} \times \displaystyle \frac{\pi}{180^{\circ}}=\frac{-275\pi}{180}=\, – \frac{55\pi}{36}\)
\(\text{Degrees } \rightarrow \text{ Radians}\)
\(\,\,\,\,\, -275^{\circ} \times \displaystyle \frac{\pi}{180^{\circ}}=\frac{-275\pi}{180}=\, – \frac{55\pi}{36}\)
\(\textbf{4)}\) Convert \( \theta=140^{\circ} \) from degrees to radians The answer is \( \displaystyle\frac{7\pi}{9} \) radians
\(\text{Degrees } \rightarrow \text{ Radians}\)
\(\,\,\,\,\, 140^{\circ} \times \displaystyle \frac{\pi}{180^{\circ}}=\frac{140\pi}{180}=\frac{7\pi}{9}\)
\(\text{Degrees } \rightarrow \text{ Radians}\)
\(\,\,\,\,\, 140^{\circ} \times \displaystyle \frac{\pi}{180^{\circ}}=\frac{140\pi}{180}=\frac{7\pi}{9}\)
\(\textbf{5)}\) Convert \( \theta=-36^{\circ} \) from degrees to radians The answer is \( – \displaystyle \frac{\pi}{5} \) radians
\(\text{Degrees } \rightarrow \text{ Radians}\)
\(\,\,\,\,\, -36^{\circ} \times \displaystyle \frac{\pi}{180^{\circ}}=\frac{-36\pi}{180}=\, – \frac{\pi}{5}\)
\(\text{Degrees } \rightarrow \text{ Radians}\)
\(\,\,\,\,\, -36^{\circ} \times \displaystyle \frac{\pi}{180^{\circ}}=\frac{-36\pi}{180}=\, – \frac{\pi}{5}\)
\(\textbf{6)}\) Convert \( \theta=400^{\circ} \) from degrees to radians The answer is \( \displaystyle\frac{20\pi}{9} \) radians
\(\text{Degrees } \rightarrow \text{ Radians}\)
\(\,\,\,\,\, 400^{\circ} \times \displaystyle \frac{\pi}{180^{\circ}}=\frac{400\pi}{180}=\frac{20\pi}{9}\)
\(\text{Degrees } \rightarrow \text{ Radians}\)
\(\,\,\,\,\, 400^{\circ} \times \displaystyle \frac{\pi}{180^{\circ}}=\frac{400\pi}{180}=\frac{20\pi}{9}\)
\(\textbf{7)}\) Convert \( x=\displaystyle\frac{-7\pi}{5} \) from radians to degrees The answer is \( -252^{\circ} \)
\(\text{Radians } \rightarrow \text{ Degrees}\)
\(\,\,\,\,\, \displaystyle\frac{-7\pi}{5} \times \displaystyle \frac{180^{\circ}}{\pi}=-252^{\circ}\)
\(\text{Radians } \rightarrow \text{ Degrees}\)
\(\,\,\,\,\, \displaystyle\frac{-7\pi}{5} \times \displaystyle \frac{180^{\circ}}{\pi}=-252^{\circ}\)
\(\textbf{8)}\) Convert \( x=\displaystyle\frac{-5\pi}{3} \) from radians to degrees The answer is \( -300^{\circ} \)
\(\text{Radians } \rightarrow \text{ Degrees}\)
\(\,\,\,\,\, \displaystyle\frac{-5\pi}{3} \times \displaystyle \frac{180^{\circ}}{\pi}=-300^{\circ}\)
\(\text{Radians } \rightarrow \text{ Degrees}\)
\(\,\,\,\,\, \displaystyle\frac{-5\pi}{3} \times \displaystyle \frac{180^{\circ}}{\pi}=-300^{\circ}\)
\(\textbf{9)}\) Convert \( x=\displaystyle\frac{4\pi}{5} \) from radians to degrees The answer is \( 144^{\circ} \)
\(\text{Radians } \rightarrow \text{ Degrees}\)
\(\,\,\,\,\, \displaystyle\frac{4\pi}{5} \times \displaystyle \frac{180^{\circ}}{\pi}=144^{\circ}\)
\(\text{Radians } \rightarrow \text{ Degrees}\)
\(\,\,\,\,\, \displaystyle\frac{4\pi}{5} \times \displaystyle \frac{180^{\circ}}{\pi}=144^{\circ}\)
\(\textbf{10)}\) Convert \( x=\displaystyle\frac{\pi}{8} \) from radians to degrees The answer is \( 22.5^{\circ} \)
\(\text{Radians } \rightarrow \text{ Degrees}\)
\(\,\,\,\,\, \displaystyle\frac{\pi}{8} \times \displaystyle \frac{180^{\circ}}{\pi}=22.5^{\circ}\)
\(\text{Radians } \rightarrow \text{ Degrees}\)
\(\,\,\,\,\, \displaystyle\frac{\pi}{8} \times \displaystyle \frac{180^{\circ}}{\pi}=22.5^{\circ}\)
\(\textbf{11)}\) Convert \( x=\displaystyle\frac{5\pi}{12} \) from radians to degrees The answer is \( 75^{\circ} \)
\(\text{Radians } \rightarrow \text{ Degrees}\)
\(\,\,\,\,\, \displaystyle\frac{5\pi}{12} \times \displaystyle \frac{180^{\circ}}{\pi}=75^{\circ}\)
\(\text{Radians } \rightarrow \text{ Degrees}\)
\(\,\,\,\,\, \displaystyle\frac{5\pi}{12} \times \displaystyle \frac{180^{\circ}}{\pi}=75^{\circ}\)
\(\textbf{12)}\) Convert \( \theta=315^{\circ} \) from degrees to radians The answer is \(\displaystyle\frac{7\pi}{4}\) radians
\(\text{Degrees } \rightarrow \text{ Radians}\)
\(\,\,\,\,\,315^{\circ}\times\displaystyle\frac{\pi}{180^{\circ}}=\frac{315\pi}{180}\)
\(\,\,\,\,\,\frac{315\pi}{180}=\frac{7\pi}{4}\)
\(\text{Degrees } \rightarrow \text{ Radians}\)
\(\,\,\,\,\,315^{\circ}\times\displaystyle\frac{\pi}{180^{\circ}}=\frac{315\pi}{180}\)
\(\,\,\,\,\,\frac{315\pi}{180}=\frac{7\pi}{4}\)
\(\textbf{13)}\) Convert \( \theta=225^{\circ} \) from degrees to radians The answer is \(\displaystyle\frac{5\pi}{4}\) radians
\(\text{Degrees } \rightarrow \text{ Radians}\)
\(\,\,\,\,\,225^{\circ}\times\displaystyle\frac{\pi}{180^{\circ}}=\frac{225\pi}{180}\)
\(\,\,\,\,\,\frac{225\pi}{180}=\frac{5\pi}{4}\)
\(\text{Degrees } \rightarrow \text{ Radians}\)
\(\,\,\,\,\,225^{\circ}\times\displaystyle\frac{\pi}{180^{\circ}}=\frac{225\pi}{180}\)
\(\,\,\,\,\,\frac{225\pi}{180}=\frac{5\pi}{4}\)
\(\textbf{14)}\) Convert \( \theta=-150^{\circ} \) from degrees to radians The answer is \(-\displaystyle\frac{5\pi}{6}\) radians
\(\text{Degrees } \rightarrow \text{ Radians}\)
\(\,\,\,\,\,-150^{\circ}\times\displaystyle\frac{\pi}{180^{\circ}}=\frac{-150\pi}{180}\)
\(\,\,\,\,\,\frac{-150\pi}{180}=-\frac{5\pi}{6}\)
\(\text{Degrees } \rightarrow \text{ Radians}\)
\(\,\,\,\,\,-150^{\circ}\times\displaystyle\frac{\pi}{180^{\circ}}=\frac{-150\pi}{180}\)
\(\,\,\,\,\,\frac{-150\pi}{180}=-\frac{5\pi}{6}\)
\(\textbf{15)}\) Convert \( x=\displaystyle\frac{11\pi}{6} \) from radians to degrees The answer is \(330^{\circ}\)
\(\text{Radians } \rightarrow \text{ Degrees}\)
\(\,\,\,\,\,\displaystyle\frac{11\pi}{6}\times\displaystyle\frac{180^{\circ}}{\pi}\)
\(\,\,\,\,\,\displaystyle\frac{11\cdot180^{\circ}}{6}=330^{\circ}\)
\(\text{Radians } \rightarrow \text{ Degrees}\)
\(\,\,\,\,\,\displaystyle\frac{11\pi}{6}\times\displaystyle\frac{180^{\circ}}{\pi}\)
\(\,\,\,\,\,\displaystyle\frac{11\cdot180^{\circ}}{6}=330^{\circ}\)
Challenge Problems
\(\textbf{16)}\) Convert \( x=-\displaystyle\frac{13\pi}{4} \) from radians to degrees The answer is \(-585^{\circ}\)
\(\text{Radians } \rightarrow \text{ Degrees}\)
\(\,\,\,\,\,-\displaystyle\frac{13\pi}{4}\times\displaystyle\frac{180^{\circ}}{\pi}\)
\(\,\,\,\,\,-\displaystyle\frac{13\cdot180^{\circ}}{4}=-13\cdot45^{\circ}\)
\(\,\,\,\,\,-13\cdot45^{\circ}=-585^{\circ}\)
\(\text{Radians } \rightarrow \text{ Degrees}\)
\(\,\,\,\,\,-\displaystyle\frac{13\pi}{4}\times\displaystyle\frac{180^{\circ}}{\pi}\)
\(\,\,\,\,\,-\displaystyle\frac{13\cdot180^{\circ}}{4}=-13\cdot45^{\circ}\)
\(\,\,\,\,\,-13\cdot45^{\circ}=-585^{\circ}\)
\(\textbf{17)}\) Convert \( \theta=765^{\circ} \) from degrees to radians The answer is \(\displaystyle\frac{17\pi}{4}\) radians
\(\text{Degrees } \rightarrow \text{ Radians}\)
\(\,\,\,\,\,765^{\circ}\times\displaystyle\frac{\pi}{180^{\circ}}=\frac{765\pi}{180}\)
\(\,\,\,\,\,\frac{765\pi}{180}=\frac{17\pi}{4}\)
\(\text{Degrees } \rightarrow \text{ Radians}\)
\(\,\,\,\,\,765^{\circ}\times\displaystyle\frac{\pi}{180^{\circ}}=\frac{765\pi}{180}\)
\(\,\,\,\,\,\frac{765\pi}{180}=\frac{17\pi}{4}\)
\(\textbf{18)}\) Convert \( x=\displaystyle\frac{19\pi}{10} \) from radians to degrees The answer is \(342^{\circ}\)
\(\text{Radians } \rightarrow \text{ Degrees}\)
\(\,\,\,\,\,\displaystyle\frac{19\pi}{10}\times\displaystyle\frac{180^{\circ}}{\pi}\)
\(\,\,\,\,\,\displaystyle\frac{19\cdot180^{\circ}}{10}=19\cdot18^{\circ}\)
\(\,\,\,\,\,19\cdot18^{\circ}=342^{\circ}\)
\(\text{Radians } \rightarrow \text{ Degrees}\)
\(\,\,\,\,\,\displaystyle\frac{19\pi}{10}\times\displaystyle\frac{180^{\circ}}{\pi}\)
\(\,\,\,\,\,\displaystyle\frac{19\cdot180^{\circ}}{10}=19\cdot18^{\circ}\)
\(\,\,\,\,\,19\cdot18^{\circ}=342^{\circ}\)
\(\textbf{19)}\) Convert \( \theta=-720^{\circ} \) from degrees to radians The answer is \(-4\pi\) radians
\(\text{Degrees } \rightarrow \text{ Radians}\)
\(\,\,\,\,\,-720^{\circ}\times\displaystyle\frac{\pi}{180^{\circ}}=\frac{-720\pi}{180}\)
\(\,\,\,\,\,\frac{-720\pi}{180}=-4\pi\)
\(\text{Degrees } \rightarrow \text{ Radians}\)
\(\,\,\,\,\,-720^{\circ}\times\displaystyle\frac{\pi}{180^{\circ}}=\frac{-720\pi}{180}\)
\(\,\,\,\,\,\frac{-720\pi}{180}=-4\pi\)
\(\textbf{20)}\) Convert \( x=-\displaystyle\frac{17\pi}{12} \) from radians to degrees The answer is \(-255^{\circ}\)
\(\text{Radians } \rightarrow \text{ Degrees}\)
\(\,\,\,\,\,-\displaystyle\frac{17\pi}{12}\times\displaystyle\frac{180^{\circ}}{\pi}\)
\(\,\,\,\,\,-\displaystyle\frac{17\cdot180^{\circ}}{12}=-17\cdot15^{\circ}\)
\(\,\,\,\,\,-17\cdot15^{\circ}=-255^{\circ}\)
\(\text{Radians } \rightarrow \text{ Degrees}\)
\(\,\,\,\,\,-\displaystyle\frac{17\pi}{12}\times\displaystyle\frac{180^{\circ}}{\pi}\)
\(\,\,\,\,\,-\displaystyle\frac{17\cdot180^{\circ}}{12}=-17\cdot15^{\circ}\)
\(\,\,\,\,\,-17\cdot15^{\circ}=-255^{\circ}\)
See Related Pages\(\)
\(\bullet\text{ Visualizing Radians}\)
\(\,\,\,\,\,\,\,\,\text{(Geogebra.org)}\)
\(\bullet\text{ Right Triangle Trigonometry}\)
\(\,\,\,\,\,\,\,\,\sin{(x)}=\displaystyle\frac{\text{opp}}{\text{hyp}}…\)
\(\bullet\text{ Angle of Depression and Elevation}\)
\(\,\,\,\,\,\,\,\,\text{Angle of Depression}=\text{Angle of Elevation}…\)
\(\bullet\text{ Convert to Radians and to Degrees}\)
\(\,\,\,\,\,\,\,\,\text{Radians} \rightarrow \text{Degrees}, \times \displaystyle \frac{180^{\circ}}{\pi}…\)
\(\bullet\text{ Degrees, Minutes and Seconds}\)
\(\,\,\,\,\,\,\,\,48^{\circ}34’21”…\)
\(\bullet\text{ Coterminal Angles}\)
\(\,\,\,\,\,\,\,\,\pm 360^{\circ} \text { or } \pm 2\pi n…\)
\(\bullet\text{ Reference Angles}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
\(\bullet\text{ Find All 6 Trig Functions}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
\(\bullet\text{ Unit Circle}\)
\(\,\,\,\,\,\,\,\,\sin{(60^{\circ})}=\displaystyle\frac{\sqrt{3}}{2}…\)
\(\bullet\text{ Law of Sines}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{\sin{A}}{a}=\frac{\sin{B}}{b}=\frac{\sin{C}}{c}\) 
\(…\)
\(\bullet\text{ Area of SAS Triangles}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\frac{1}{2}ab \sin{C}\) \(…\)
\(\bullet\text{ Law of Cosines}\)
\(\,\,\,\,\,\,\,\,a^2=b^2+c^2-2bc \cos{A}\) \(…\)
\(\bullet\text{ Area of SSS Triangles (Heron’s formula)}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\sqrt{s(s-a)(s-b)(s-c)}\) \(…\)
\(\bullet\text{ Geometric Mean}\)
\(\,\,\,\,\,\,\,\,x=\sqrt{ab} \text{ or } \displaystyle\frac{a}{x}=\frac{x}{b}…\)
\(\bullet\text{ Geometric Mean- Similar Right Triangles}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
\(\bullet\text{ Inverse Trigonmetric Functions}\)
\(\,\,\,\,\,\,\,\,\sin {\left(cos^{-1}\left(\frac{3}{5}\right)\right)}…\)
\(\bullet\text{ Sum and Difference of Angles Formulas}\)
\(\,\,\,\,\,\,\,\,\sin{(A+B)}=\sin{A}\cos{B}+\cos{A}\sin{B}…\)
\(\bullet\text{ Double-Angle and Half-Angle Formulas}\)
\(\,\,\,\,\,\,\,\,\sin{(2A)}=2\sin{(A)}\cos{(A)}…\)
\(\bullet\text{ Trigonometry-Pythagorean Identities}\)
\(\,\,\,\,\,\,\,\,\sin^2{(x)}+\cos^2{(x)}=1…\)
\(\bullet\text{ Product-Sum Identities}\)
\(\,\,\,\,\,\,\,\,\cos{\alpha}\cos{\beta}=\left(\displaystyle\frac{\cos{(\alpha+\beta)}+\cos{(\alpha-\beta)}}{2}\right)…\)
\(\bullet\text{ Cofunction Identities}\)
\(\,\,\,\,\,\,\,\,\sin{(x)}=\cos{(\frac{\pi}{2}-x)}…\)
\(\bullet\text{ Proving Trigonometric Identities}\)
\(\,\,\,\,\,\,\,\,\sec{x}-\cos{x}=\displaystyle\frac{\tan^2{x}}{\sec{x}}…\)
\(\bullet\text{ Graphing Trig Functions- sin and cos}\)
\(\,\,\,\,\,\,\,\,f(x)=A \sin{B(x-c)}+D \) 
\(…\)
\(\bullet\text{ Solving Trigonometric Equations}\)
\(\,\,\,\,\,\,\,\,2\cos{(x)}=\sqrt{3}…\)