Practice Problems

Solve each trigonometric equation in radians or degrees.

\(\textbf{1)}\) \(2 \cos(x)=\sqrt{3}\)

Show Answer

\(x=-\frac{1}{6}\pi+2\pi\text{n}, \,\,\text{or}\,\, \frac{1}{6}\pi+2\pi\text{n}\)
\(x=-30^{\circ}+360\text{n}^{\circ}, \,\,\text{or}\,\, 30^{\circ}+360\text{n}^{\circ}\)

Show Work

\(\,\,\,\,\,\,2 \cos(x)=\sqrt{3}\)
\(\,\,\,\,\,\,\cos(x)=\frac{\sqrt{3}}{2}\)
\(\,\,\,\,\,\,\cos^{-1}\left(\cos(x)\right)=\cos^{-1}\left(\frac{\sqrt{3}}{2}\right)\)
\(\,\,\,\,\,\,x=-\frac{1}{6}\pi+2\pi\text{n}, \,\,\text{or}\,\, \frac{1}{6}\pi+2\pi\text{n}\)
\(\,\,\,\,\,\,x=-30^{\circ}+360\text{n}^{\circ}, \,\,\text{or}\,\, 30^{\circ}+360\text{n}^{\circ}\)

 

\(\textbf{2)}\) \(2\tan^2{\theta}=1+\tan^2{\theta}, \,\,\, 0^{\circ}\le\theta\lt360^{\circ}\)

Show Answer

\(\theta=45^{\circ}, 135^{\circ}, 225^{\circ}, 315^{\circ}\)

Show Work

\(\,\,\,\,\,\,2\tan^2{\theta}=1+\tan^2{\theta}\)
\(\,\,\,\,\,\,\tan^2{\theta}=1\)
\(\,\,\,\,\,\,\sqrt{\tan^2{\theta}}=\pm\sqrt{1}\)
\(\,\,\,\,\,\,\tan{\theta}=\pm1\)
\(\,\,\,\,\,\,\tan{\theta}=-1\)\( \,\,\text{OR}\,\,\)\( \tan{\theta}=1\)
\(\,\,\,\,\,\,\tan^{-1}\left(\tan{\theta}\right)=\tan^{-1}\left(-1\right)\)\( \,\,\text{OR}\,\,\)\(\tan^{-1}\left(\tan{\theta}\right)=\tan^{-1}\left(1\right)\)
\(\,\,\,\,\,\,\theta=135^{\circ}, 225^{\circ} \)\( \,\,\text{OR}\,\,\)\( \theta= 45^{\circ}, 315^{\circ}\)

\(\,\,\,\,\,\,\theta=45^{\circ}, 135^{\circ}, 225^{\circ}, 315^{\circ}\)

 

\(\textbf{3)}\) \(8\sin^2{\theta}-2=0, \,\,\, 0^{\circ}\le\theta\lt360^{\circ}\)

Show Answer

\(\theta=30^{\circ}, 150^{\circ}, 210^{\circ}, 330^{\circ}\)

Show Work

\(\,\,\,\,\,\,8\sin^2{\theta}-2=0\)
\(\,\,\,\,\,\,8\sin^2{\theta}=2\)
\(\,\,\,\,\,\,\sin^2{\theta}=\frac{1}{4}\)
\(\,\,\,\,\,\,\sqrt{\sin^2{\theta}}=\pm\sqrt{\frac{1}{4}}\)
\(\,\,\,\,\,\,\sin{\theta}=\pm\frac{1}{2}\)
\(\,\,\,\,\,\,\sin{\theta}=-\frac{1}{2}\)\( \,\,\text{OR}\,\,\)\( \sin{\theta}=\frac{1}{2}\)
\(\,\,\,\,\,\,\sin^{-1}\left(\sin{\theta}\right)=\sin^{-1}\left(-\frac{1}{2}\right)\)\( \,\,\text{OR}\,\,\)\( \sin^{-1}\left(\sin{\theta}\right)=\sin^{-1}\left(\frac{1}{2}\right)\)
\(\,\,\,\,\,\,\theta=210^{\circ}, 330^{\circ} \)\( \,\,\text{OR}\,\,\)\( \theta= 30^{\circ}, 150^{\circ}\)

\(\,\,\,\,\,\,\theta=30^{\circ}, 150^{\circ}, 210^{\circ}, 330^{\circ}\)

 

\(\textbf{4)}\) \(2\sin^2{\theta}+3\sin{\theta}+1=0, \,\,\, 0^{\circ}\le\theta\lt360^{\circ}\)

Show Answer

\(\theta=210^{\circ}, 270^{\circ}, 330^{\circ},\)

Show Work

\(\,\,\,\,\,\,2\sin^2{\theta}+3\sin{\theta}+1=0\)
\(\,\,\,\,\,\,u=\sin{\theta}\)
\(\,\,\,\,\,\,2u^2+3u+1=0\)
\(\,\,\,\,\,\,\left(2u+1\right)\left(u+1\right)=0\)
\(\,\,\,\,\,\,\left(2\sin{\theta}+1\right)\left(\sin{\theta}+1\right)=0\)
\(\,\,\,\,\,\,2\sin{\theta}+1=0\)\( \,\,\text{OR}\,\,\)\( \sin{\theta}+1=0\)
\(\,\,\,\,\,\,2\sin{\theta}=-1\)\( \,\,\text{OR}\,\,\)\( \sin{\theta}=-1\)
\(\,\,\,\,\,\,\sin{\theta}=-\frac{1}{2}\)\( \,\,\text{OR}\,\,\)\( \sin{\theta}=-1\)
\(\,\,\,\,\,\,\sin^{-1}\left(\sin{\theta}\right)=\sin^{-1}\left(-\frac{1}{2}\right)\)\( \,\,\text{OR}\,\,\)\( \sin^{-1}\left(\sin{\theta}\right)=\sin^{-1}\left(-1\right)\)
\(\,\,\,\,\,\,\theta=210^{\circ}, 330^{\circ} \)\( \,\,\text{OR}\,\,\)\( \theta= 270^{\circ}\)

\(\,\,\,\,\,\,\theta=210^{\circ}, 270^{\circ}, 330^{\circ}\)

 

\(\textbf{5)}\) \(\sqrt{3}\csc{x}-2=0, \,\,\, 0^{\circ}\le\theta\lt360^{\circ}\)

Show Answer

\(\theta= 60^{\circ}, 120^{\circ}\)

Show Work

\(\,\,\,\,\,\,\sqrt{3}\csc{x}-2=0\)
\(\,\,\,\,\,\,\sqrt{3}\csc{x}=2\)
\(\,\,\,\,\,\,\csc{x}=\frac{2}{\sqrt{3}}\)
\(\,\,\,\,\,\,\sin{x}=\frac{\sqrt{3}}{2}\)
\(\,\,\,\,\,\,\sin^{-1}\left(\sin{x}\right)=\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\)

\(\,\,\,\,\,\,\theta=60^{\circ}, 120^{\circ}\)

 

\(\textbf{6)}\) \(2 \sin(4x)=\sqrt{3}\)

Show Answer

\(x=\frac{1}{6}\pi+\frac{1}{2}\pi\text{n}, \,\,\text{or}\,\, \frac{1}{12}\pi+\frac{1}{2}\pi\text{n}\)
\(x=30^{\circ}+90\text{n}^{\circ}, \,\,\text{or}\,\, 15^{\circ}+90\text{n}^{\circ}\)

Show Work

\(\,\,\,\,\,\,2 \sin(4x)=\sqrt{3}\)
\(\,\,\,\,\,\,\sin(4x)=\frac{\sqrt{3}}{2}\)
\(\,\,\,\,\,\,\sin^{-1}\left(\sin(4x)\right)=\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\)
\(\,\,\,\,\,\,4x=\frac{2}{3}\pi+2\pi\text{n}, \,\,\text{or}\,\, 4x=\frac{1}{3}\pi+2\pi\text{n}\)
\(\,\,\,\,\,\,x=\frac{1}{6}\pi+\frac{1}{2}\pi\text{n}, \,\,\text{or}\,\, x=\frac{1}{12}\pi+\frac{1}{2}\pi\text{n}\)

 

\(\textbf{7)}\) \(2 \sin(3x)=\sqrt{3}\)

Show Answer

\(x=\frac{1}{9}\pi+\frac{2}{3}\pi\text{n}, \,\,\text{or}\,\, \frac{2}{9}\pi+\frac{2}{3}\pi\text{n}\)
\(x=20^{\circ}+120\text{n}^{\circ}, \,\,\text{or}\,\, 40^{\circ}+120\text{n}^{\circ}\)

Show Work

\(\,\,\,\,\,\,2 \sin(3x)=\sqrt{3}\)
\(\,\,\,\,\,\,\sin(3x)=\frac{\sqrt{3}}{2}\)
\(\,\,\,\,\,\,\sin^{-1}\left(\sin(3x)\right)=\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\)
\(\,\,\,\,\,\,3x=\frac{1}{3}\pi+2\pi\text{n}, \,\,\text{or}\,\, x=\frac{2}{3}\pi+2\pi\text{n}\)
\(\,\,\,\,\,\,x=\frac{1}{9}\pi+\frac{2}{3}\pi\text{n}, \,\,\text{or}\,\, x=\frac{2}{9}\pi+\frac{2}{3}\pi\text{n}\)

 

\(\textbf{8)}\) \( \cos(2x)=1\)

Show Answer

\(x=\pi\text{n}\)
\(x=180\text{n}^{\circ}\)

Show Work

\(\,\,\,\,\,\,\cos(2x)=1\)
\(\,\,\,\,\,\,\cos^{-1}\left(\cos(2x)\right)=\cos^{-1}(1)\)
\(\,\,\,\,\,\,2x=2\pi\text{n}\)
\(\,\,\,\,\,\,x=\pi\text{n}\)

 

\(\textbf{9)}\) \( \cos(x)+2=1\)

Show Answer

\(x=\pi+ 2\pi\text{n}\)
\(x=180^{\circ}+ 360\text{n}^{\circ}\)

Show Work

\(\,\,\,\,\,\,\cos(x)+2=1\)
\(\,\,\,\,\,\,\cos(x)=-1\)
\(\,\,\,\,\,\,\cos^{-1}\left(\cos(x)\right)=\cos^{-1}(-1)\)
\(\,\,\,\,\,\,x=\pi+2\pi\text{n}\)

 

\(\textbf{10)}\) \( \sin(x)=\cos(x)\)

Show Answer

\(x=\frac{1}{4}\pi+\pi\text{n}\)
\(x=45^{\circ}+180\text{n}^{\circ}\)

Show Work

\(\,\,\,\,\,\,\sin(x)=\cos(x)\)
\(\,\,\,\,\,\,\displaystyle\frac{\sin(x)}{\cos(x)}=\frac{\cos(x)}{\cos(x)}\)
\(\,\,\,\,\,\,\tan(x)=1\)
\(\,\,\,\,\,\,\tan^{-1}\left(\tan(x)\right)=\tan^{-1}(1)\)
\(\,\,\,\,\,\,x=\frac{1}{4}\pi+\pi\text{n}\)

 

\(\textbf{11)}\) \( \tan^2(x)+3=0\)

Show Answer

No solution

Show Work

\(\,\,\,\,\,\,\tan^2(x)+3=0\)
\(\,\,\,\,\,\,\tan^2(x)=-3\)
\(\,\,\,\,\,\,\sqrt{\tan^2(x)}=\sqrt{-3}\)
\(\,\,\,\,\,\,\)No Solution

 

\(\textbf{12)}\) \( 2\cos^2(x)+\sqrt{3}\cos(x)=0\)

Show Answer

\(x=-\frac{1}{2}\pi+\pi\text{n}, \,\,\text{or}\,\, -\frac{5}{6}\pi+2\pi\text{n}, \,\,\text{or}\,\, \frac{5}{6}\pi+2\pi\text{n}\)
\(x=-90^{\circ}+180\text{n}^{\circ}, \,\,\text{or}\,\, -150^{\circ}+360\text{n}^{\circ}, \,\,\text{or}\,\, 150^{\circ}+360\text{n}^{\circ}\)

Show Work

\(\,\,\,\,\,\,2\cos^2(x)+\sqrt{3}\cos(x)=0\)
\(\,\,\,\,\,\,\left(\cos(x)\right)\left(2\cos(x)+\sqrt{3}\right)=0\)
\(\,\,\,\,\,\,\cos(x)=0\) or \(2\cos(x)+\sqrt{3}=0\)
\(\,\,\,\,\,\,\cos(x)=0\) or \(2\cos(x)=-\sqrt{3}\)
\(\,\,\,\,\,\,\cos(x)=0\) or \(\cos(x)=-\frac{\sqrt{3}}{2}\)
\(\,\,\,\,\,\,\cos^{-1}\left(\cos(x)\right)=\cos^{-1}(0)\) or \(\cos^{-1}\left(\cos(x)\right)=\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right)\)
\(\,\,\,\,\,\,x=-\frac{1}{2}\pi+\pi\text{n}, \,\,\text{or}\,\, -\frac{5}{6}\pi+2\pi\text{n}, \,\,\text{or}\,\, \frac{5}{6}\pi+2\pi\text{n}\)

 

\(\textbf{13)}\) \( \sin^2(x)+\sin(x)=2\)

Show Answer

\(x=\frac{1}{2}\pi+2\pi\text{n}\)
\(x=90^{\circ}+360\text{n}^{\circ}\)

Show Work

\(\,\,\,\,\,\,\sin^2(x)+\sin(x)-2=0\)
\(\,\,\,\,\,\,\left(\sin(x)-2\right)\left(\sin(x)+1\right)\)
\(\,\,\,\,\,\,\sin(x)-2=0\) or \(\sin(x)+1=0\)
\(\,\,\,\,\,\,\sin(x)=2\) or \(\sin(x)=-1\)
\(\,\,\,\,\,\,\sin^{-1}\left(\sin(x)\right)=\sin^{-1}(2)\) or \(\sin^{-1}\left(\sin(x)\right)=\sin^{-1}(-1)\)
\(\,\,\,\,\,\,\) No Solution or \(x=\frac{1}{2}\pi+2\pi\text{n}\)
\(\,\,\,\,\,\,x=\frac{1}{2}\pi+2\pi\text{n}\)

 

\(\textbf{14)}\) \( \sin^2(x)+\sin(x)=\cos^2(x)\)

Show Answer

\(x=-\frac{1}{2}\pi+2\pi\text{n}, \,\,\text{or}\,\, \frac{1}{6}\pi+2\pi\text{n}, \,\,\text{or}\,\, \frac{5}{6}\pi+2\pi\text{n}\)
\(x=-90^{\circ}+360\text{n}^{\circ}, \,\,\text{or}\,\, 30^{\circ}+360\text{n}^{\circ}, \,\,\text{or}\,\, 150^{\circ}+360\text{n}^{\circ}\)

Show Work

\(\,\,\,\,\,\,\sin^2(x)+\sin(x)=\cos^2(x)\)
\(\,\,\,\,\,\,\sin^2(x)+\sin(x)=1-\sin^2(x) \,\, \left(\text{Pythagorean Identities}\right)\)
\(\,\,\,\,\,\,2\sin^2(x)+\sin(x)-1=0\)
\(\,\,\,\,\,\,\left(2\sin(x)+1\right)\left(\sin(x)-1\right)\)
\(\,\,\,\,\,\,2\sin(x)+1=0\) or \(\sin(x)-1=0\)
\(\,\,\,\,\,\,2\sin(x)=-1\) or \(\sin(x)=1\)
\(\,\,\,\,\,\,\sin(x)=-\frac{1}{2}\) or \(\sin(x)=1\)
\(\,\,\,\,\,\,\sin^{-1}\left(\sin(x)\right)=\sin^{-1}\left(-\frac{1}{2}\right)\) or \(\sin^{-1}\left(\sin(x)\right)=\sin^{-1}(1)\)
\(\,\,\,\,\,\,x=-\frac{1}{2}\pi+2\pi\text{n}, \,\,\text{or}\,\, \frac{1}{6}\pi+2\pi\text{n}, \,\,\text{or}\,\, \frac{5}{6}\pi+2\pi\text{n}\)

 

\(\textbf{15)}\) \( \cos(2x)=\cos(x)\)

Show Answer

\(x=2\pi\text{n}, \,\,\text{or}\,\, -\frac{2}{3}\pi+2\pi\text{n}, \,\,\text{or}\,\, \frac{2}{3}\pi+2\pi\text{n}\)
\(x=360\text{n}^{\circ}, \,\,\text{or}\,\, -120^{\circ}+360\text{n}^{\circ}, \,\,\text{or}\,\, 120^{\circ}+360\text{n}^{\circ}\)

 

\(\textbf{16)}\) \( \sin(x)+\cos(x)=1\)

Show Answer

\(x=2\pi\text{n}, \,\,\text{or}\,\, \frac{1}{2}\pi+2\pi\text{n}\)
\(x=360\text{n}^{\circ}, \,\,\text{or}\,\, 90^{\circ}+360\text{n}^{\circ}\)

 

\(\textbf{17)}\) \( \tan(x) – 2\sin(x) = 0\)

Show Answer

\(x=\pi\text{n}, \,\,\text{or}\,\, -\frac{1}{3}\pi+2\pi\text{n}, \,\,\text{or}\,\, \frac{1}{3}\pi+2\pi\text{n}\)
\(x=180\text{n}^{\circ}, \,\,\text{or}\,\, -60^{\circ}+360\text{n}^{\circ}, \,\,\text{or}\,\, 60^{\circ}+360\text{n}^{\circ}\)

 

\(\textbf{18)}\) \( 2\sin(4x)=1\)

Show Answer

\(x=\frac{1}{24}\pi+\frac{1}{2}\pi\text{n}, \,\,\text{or}\,\, \frac{5}{24}\pi+\frac{1}{2}\pi\text{n}\)
\(x=7.5^{\circ}+90\text{n}^{\circ}, \,\,\text{or}\,\, 37.5^{\circ}+90\text{n}^{\circ}\)

 

\(\textbf{19)}\) \( \sin(x)+\cos(x)=2\)

Show Answer

No solution

 

\(\textbf{20)}\) \( \sin(x)+\sqrt{3}\cos(x)=0\)

Show Answer

\(x=\displaystyle \frac{2\pi}{3}+\pi n\)
\(x=120^{\circ \:}+180^{\circ \:}n\)

 

\(\textbf{21)}\) \(2\cos^2(2x)=1\)

Show Answer

\(x=\frac{\pi}{8}+\pi n, \,\, x=\pi-\frac{\pi}{8}+\pi n, \,\, x=\frac{3\pi}{8}+\pi n, \,\, x=-\frac{3\pi}{8}+\pi n\)

Show Work

\(\,\,\,\,\,\,2\cos^2(2x)=1\)
\(\,\,\,\,\,\,\cos^2(2x)=\tfrac{1}{2}\)
\(\,\,\,\,\,\,\cos(2x)=\pm\tfrac{\sqrt{2}}{2}\)
\(\,\,\,\,\,\,2x=\pm\tfrac{\pi}{4}+2\pi n, \,\,\,\, 2x=\pi\pm\tfrac{\pi}{4}+2\pi n\)
\(\,\,\,\,\,\,x=\tfrac{\pi}{8}+\pi n, \,\, x=\pi-\tfrac{\pi}{8}+\pi n, \,\, x=\tfrac{3\pi}{8}+\pi n, \,\, x=-\tfrac{3\pi}{8}+\pi n\)

 

Challenge Questions

\(\textbf{22)}\) \( \sin^{-1}(\sqrt{3x})=\cos^{-1}(\sqrt{x})\)

Show Answer

The answer is \(x=\frac{1}{4}\)

 

See Related Pages\(\)

\(\bullet\text{ Right Triangle Trigonometry}\)
\(\,\,\,\,\,\,\,\,\sin{(x)}=\displaystyle\frac{\text{opp}}{\text{hyp}}…\)

\(\bullet\text{ Angle of Depression and Elevation}\)
\(\,\,\,\,\,\,\,\,\text{Angle of Depression}=\text{Angle of Elevation}…\)

\(\bullet\text{ Convert to Radians and to Degrees}\)
\(\,\,\,\,\,\,\,\,\text{Radians} \rightarrow \text{Degrees}, \times \displaystyle \frac{180^{\circ}}{\pi}…\)

\(\bullet\text{ Degrees, Minutes and Seconds}\)
\(\,\,\,\,\,\,\,\,48^{\circ}34’21”…\)

\(\bullet\text{ Coterminal Angles}\)
\(\,\,\,\,\,\,\,\,\pm 360^{\circ} \text { or } \pm 2\pi n…\)

\(\bullet\text{ Reference Angles}\)
\(\,\,\,\,\,\,\,\,\)\(…\)

\(\bullet\text{ Find All 6 Trig Functions}\)
\(\,\,\,\,\,\,\,\,\)\(…\)

\(\bullet\text{ Unit Circle}\)
\(\,\,\,\,\,\,\,\,\sin{(60^{\circ})}=\displaystyle\frac{\sqrt{3}}{2}…\)

\(\bullet\text{ Law of Sines}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{\sin{A}}{a}=\frac{\sin{B}}{b}=\frac{\sin{C}}{c}\) \(…\)

\(\bullet\text{ Area of SAS Triangles}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\frac{1}{2}ab \sin{C}\) \(…\)

\(\bullet\text{ Law of Cosines}\)
\(\,\,\,\,\,\,\,\,a^2=b^2+c^2-2bc \cos{A}\) \(…\)

\(\bullet\text{ Area of SSS Triangles (Heron’s formula)}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\sqrt{s(s-a)(s-b)(s-c)}\) \(…\)

\(\bullet\text{ Geometric Mean}\)
\(\,\,\,\,\,\,\,\,x=\sqrt{ab} \text{ or } \displaystyle\frac{a}{x}=\frac{x}{b}…\)

\(\bullet\text{ Geometric Mean- Similar Right Triangles}\)
\(\,\,\,\,\,\,\,\,\)\(…\)

\(\bullet\text{ Inverse Trigonmetric Functions}\)
\(\,\,\,\,\,\,\,\,\sin {\left(cos^{-1}\left(\frac{3}{5}\right)\right)}…\)

\(\bullet\text{ Sum and Difference of Angles Formulas}\)
\(\,\,\,\,\,\,\,\,\sin{(A+B)}=\sin{A}\cos{B}+\cos{A}\sin{B}…\)

\(\bullet\text{ Double-Angle and Half-Angle Formulas}\)
\(\,\,\,\,\,\,\,\,\sin{(2A)}=2\sin{(A)}\cos{(A)}…\)

\(\bullet\text{ Trigonometry-Pythagorean Identities}\)
\(\,\,\,\,\,\,\,\,\sin^2{(x)}+\cos^2{(x)}=1…\)

\(\bullet\text{ Product-Sum Identities}\)
\(\,\,\,\,\,\,\,\,\cos{\alpha}\cos{\beta}=\left(\displaystyle\frac{\cos{(\alpha+\beta)}+\cos{(\alpha-\beta)}}{2}\right)…\)

\(\bullet\text{ Cofunction Identities}\)
\(\,\,\,\,\,\,\,\,\sin{(x)}=\cos{(\frac{\pi}{2}-x)}…\)

\(\bullet\text{ Proving Trigonometric Identities}\)
\(\,\,\,\,\,\,\,\,\sec{x}-\cos{x}=\displaystyle\frac{\tan^2{x}}{\sec{x}}…\)

\(\bullet\text{ Graphing Trig Functions- sin and cos}\)
\(\,\,\,\,\,\,\,\,f(x)=A \sin{B(x-c)}+D \) \(…\)

\(\bullet\text{ Solving Trigonometric Equations}\)
\(\,\,\,\,\,\,\,\,2\cos{(x)}=\sqrt{3}…\)