$\mathbf{\text{5)}}$ $\sqrt{3}\csc x-2=0,0^{\circ }\le \theta <{360}^{\circ }$

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$\theta ={60}^{\circ },{120}^{\circ }$

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$\sqrt{3}\csc x-2=0$

$\sqrt{3}\csc x=2$

$\csc x=\frac{2}{\sqrt{3}}$

$\sin x=\frac{\sqrt{3}}{2}$

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



$\theta ={60}^{\circ },{120}^{\circ }$

$\mathbf{\text{6)}}$ $2\sin (4x)=\sqrt{3}$

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

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

$\sin (4x)=\frac{\sqrt{3}}{2}$

${\sin }^{-1}(\sin (4x))={\sin }^{-1}\left(\frac{\sqrt{3}}{2}\right)$

$4x=\frac{2}{3}\pi +2\pi \text{n},\text{or}4x=\frac{1}{3}\pi +2\pi \text{n}$

$x=\frac{1}{6}\pi +\frac{1}{2}\pi \text{n},\text{or}x=\frac{1}{12}\pi +\frac{1}{2}\pi \text{n}$

$\mathbf{\text{7)}}$ $2\sin (3x)=\sqrt{3}$

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

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$2\sin (3x)=\sqrt{3}$

$\sin (3x)=\frac{\sqrt{3}}{2}$

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

$3x=\frac{1}{3}\pi +2\pi \text{n},\text{or}x=\frac{2}{3}\pi +2\pi \text{n}$

$x=\frac{1}{9}\pi +\frac{2}{3}\pi \text{n},\text{or}x=\frac{2}{9}\pi +\frac{2}{3}\pi \text{n}$

$\mathbf{\text{8)}}$ $\cos (2x)=1$

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$x=\pi \text{n}$
$x=180{\text{n}}^{\circ }$

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$\cos (2x)=1$

${\cos }^{-1}(\cos (2x))={\cos }^{-1}(1)$

$2x=2\pi \text{n}$

$x=\pi \text{n}$

$\mathbf{\text{9)}}$ $\cos (x)+2=1$

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

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$\cos (x)+2=1$

$\cos (x)=-1$

${\cos }^{-1}(\cos (x))={\cos }^{-1}(-1)$

$x=\pi +2\pi \text{n}$

$\mathbf{\text{10)}}$ $\sin (x)=\cos (x)$

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

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$\sin (x)=\cos (x)$

${\displaystyle \frac{\sin (x)}{\cos (x)}=\frac{\cos (x)}{\cos (x)}}$

$\tan (x)=1$

${\tan }^{-1}(\tan (x))={\tan }^{-1}(1)$

$x=\frac{1}{4}\pi +\pi \text{n}$

$\mathbf{\text{11)}}$ ${\tan }^2(x)+3=0$

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No solution

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

${\tan }^2(x)=-3$

$\sqrt{{\tan }^2(x)}=\sqrt{-3}$

$$No Solution

$\mathbf{\text{12)}}$ $2{\cos }^2(x)+\sqrt{3}\cos (x)=0$

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

Show Work

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

$(\cos (x))(2\cos (x)+\sqrt{3})=0$

$\cos (x)=0$ or $2\cos (x)+\sqrt{3}=0$

$\cos (x)=0$ or $2\cos (x)=-\sqrt{3}$

$\cos (x)=0$ or $\cos (x)=-\frac{\sqrt{3}}{2}$

${\cos }^{-1}(\cos (x))={\cos }^{-1}(0)$ or ${\cos }^{-1}(\cos (x))={\cos }^{-1}(-\frac{\sqrt{3}}{2})$

$x=-\frac{1}{2}\pi +\pi \text{n},\text{or}-\frac{5}{6}\pi +2\pi \text{n},\text{or}\frac{5}{6}\pi +2\pi \text{n}$

$\mathbf{\text{13)}}$ ${\sin }^2(x)+\sin (x)=2$

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

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${\sin }^2(x)+\sin (x)-2=0$

$(\sin (x)-2)(\sin (x)+1)$

$\sin (x)-2=0$ or $\sin (x)+1=0$

$\sin (x)=2$ or $\sin (x)=-1$

${\sin }^{-1}(\sin (x))={\sin }^{-1}(2)$ or ${\sin }^{-1}(\sin (x))={\sin }^{-1}(-1)$

$$ No Solution or $x=\frac{1}{2}\pi +2\pi \text{n}$

$x=\frac{1}{2}\pi +2\pi \text{n}$

$\mathbf{\text{14)}}$ ${\sin }^2(x)+\sin (x)={\cos }^2(x)$

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

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

${\sin }^2(x)+\sin (x)=1-{\sin }^2(x)\left(\text{Pythagorean Identities}\right)$

$2{\sin }^2(x)+\sin (x)-1=0$

$(2\sin (x)+1)(\sin (x)-1)$

$2\sin (x)+1=0$ or $\sin (x)-1=0$

$2\sin (x)=-1$ or $\sin (x)=1$

$\sin (x)=-\frac{1}{2}$ or $\sin (x)=1$

${\sin }^{-1}(\sin (x))={\sin }^{-1}(-\frac{1}{2})$ or ${\sin }^{-1}(\sin (x))={\sin }^{-1}(1)$

$x=-\frac{1}{2}\pi +2\pi \text{n},\text{or}\frac{1}{6}\pi +2\pi \text{n},\text{or}\frac{5}{6}\pi +2\pi \text{n}$

$\mathbf{\text{15)}}$ $\cos (2x)=\cos (x)$

Show Answer

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

$\mathbf{\text{16)}}$ $\sin (x)+\cos (x)=1$

Show Answer

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

$\mathbf{\text{17)}}$ $\tan (x)–2\sin (x)=0$

Show Answer

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

$\mathbf{\text{18)}}$ $2\sin (4x)=1$

Show Answer

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

$\mathbf{\text{19)}}$ $\sin (x)+\cos (x)=2$

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No solution

$\mathbf{\text{20)}}$ $\sin (x)+\sqrt{3}\cos (x)=0$

Show Answer

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

Challenge Questions

$\mathbf{\text{21)}}$ ${\sin }^{-1}(\sqrt{3x})={\cos }^{-1}(\sqrt{x})$

Show Answer

The answer is $x=\frac{1}{4}$

See Related Pages$$

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$\sin (x)={\displaystyle \frac{\text{opp}}{\text{hyp}}\dots }$

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$$$\dots$

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$$$\dots$

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$\bullet \text{ Law of Sines}$
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$\bullet \text{ Area of SAS Triangles}$
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$\bullet \text{ Area of SSS Triangles (Heron’s formula)}$
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$\bullet \text{ Geometric Mean- Similar Right Triangles}$
$$$\dots$

$\bullet \text{ Inverse Trigonmetric Functions}$
$\sin \left(co{s}^{-1}\left(\frac{3}{5}\right)\right)\dots$

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

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

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

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

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

$\bullet \text{ Proving Trigonometric Identities}$
$\sec x-\cos x={\displaystyle \frac{{\tan }^{2}x}{\sec x}\dots }$

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$f(x)=A\sin B(x-c)+D$ $\dots$

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

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