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Trigonometry : Radians and Conversions

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Example Questions

Example Question #1 : Radians And Conversions

Change $\frac{4\pi }{3}$ $\displaystyle \frac{4\pi }{3}$ angle to degrees.

Possible Answers:

$300^{\circ}$ $\displaystyle 300^{\circ}$

$150^{\circ}$ $\displaystyle 150^{\circ}$

$240^{\circ}$ $\displaystyle 240^{\circ}$

$180^{\circ}$ $\displaystyle 180^{\circ}$

Correct answer:

$240^{\circ}$ $\displaystyle 240^{\circ}$

Explanation:

In order to change an angle into degrees, you must multiply the radian by $\frac{180^{\circ}}{\pi}$ $\displaystyle \frac{180^{\circ}}{\pi}$.

Therefore, to solve:

$\frac{4\pi }{3} \cdot \frac{180^{\circ} }{\pi} = 240^{\circ}$ $\displaystyle \frac{4\pi }{3} \cdot \frac{180^{\circ} }{\pi} = 240^{\circ}$

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Example Question #2 : Radians And Conversions

Give $\displaystyle x$ in radians:

$\frac{x+\pi}{x-\pi}=30^{\circ}$ $\displaystyle \frac{x+\pi}{x-\pi}=30^{\circ}$

Possible Answers:

$x=\frac{\pi(\pi+6)}{\pi-6}$ $\displaystyle x=\frac{\pi(\pi+6)}{\pi-6}$

$x=\frac{\pi+6}{\pi-6}$ $\displaystyle x=\frac{\pi+6}{\pi-6}$

$x=\frac{\pi(\pi+8)}{\pi-8}$ $\displaystyle x=\frac{\pi(\pi+8)}{\pi-8}$

$x=\pi$ $\displaystyle x=\pi$

$x=\frac{\pi(\pi-6)}{\pi+6}$ $\displaystyle x=\frac{\pi(\pi-6)}{\pi+6}$

Correct answer:

$x=\frac{\pi(\pi+6)}{\pi-6}$ $\displaystyle x=\frac{\pi(\pi+6)}{\pi-6}$

Explanation:

First we need to convert degrees to radians by multiplying by $\frac{\pi}{180^{\circ}}$ $\displaystyle \frac{\pi}{180^{\circ}}$:

$30^{\circ}\times \frac{\pi}{180^{\circ}}=\frac{\pi}{6}$ $\displaystyle 30^{\circ}\times \frac{\pi}{180^{\circ}}=\frac{\pi}{6}$

Now we can write:

$\frac{x+\pi}{x-\pi}=\frac{\pi}{6}\Rightarrow 6(x+\pi)=\pi(x-\pi)$ $\displaystyle \frac{x+\pi}{x-\pi}=\frac{\pi}{6}\Rightarrow 6(x+\pi)=\pi(x-\pi)$

$\Rightarrow 6x+6\pi=x\pi-\pi^2\Rightarrow 6x-x\pi=-6\pi-\pi^2$ $\displaystyle \Rightarrow 6x+6\pi=x\pi-\pi^2\Rightarrow 6x-x\pi=-6\pi-\pi^2$

$\Rightarrow x(6-\pi)=-\pi(\pi+6)$ $\displaystyle \Rightarrow x(6-\pi)=-\pi(\pi+6)$

$\Rightarrow x=\frac{-\pi(\pi+6)}{6-\pi}=\frac{\pi(\pi+6)}{\pi-6}$ $\displaystyle \Rightarrow x=\frac{-\pi(\pi+6)}{6-\pi}=\frac{\pi(\pi+6)}{\pi-6}$

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Example Question #1 : Radians And Conversions

Give $\displaystyle x$ in radians:

$x-\pi=120^{\circ}$ $\displaystyle x-\pi=120^{\circ}$

Possible Answers:

$x=\frac{5\pi}{3}$ $\displaystyle x=\frac{5\pi}{3}$

$x=\frac{\pi}{3}$ $\displaystyle x=\frac{\pi}{3}$

$x=\frac{2\pi}{3}$ $\displaystyle x=\frac{2\pi}{3}$

$x=\frac{4\pi}{3}$ $\displaystyle x=\frac{4\pi}{3}$

$x=\frac{5\pi}{6}$ $\displaystyle x=\frac{5\pi}{6}$

Correct answer:

$x=\frac{5\pi}{3}$ $\displaystyle x=\frac{5\pi}{3}$

Explanation:

First we need to convert degrees to radians by multiplying by $\frac{\pi}{180^{\circ}}$ $\displaystyle \frac{\pi}{180^{\circ}}$:

$120^{\circ}\times \frac{\pi}{180^{\circ}}=\frac{2\pi}{3}$ $\displaystyle 120^{\circ}\times \frac{\pi}{180^{\circ}}=\frac{2\pi}{3}$

Now we can write:

$x-\pi=120^{\circ}\Rightarrow x-\pi=\frac{2\pi}{3}\Rightarrow x=\frac{2\pi}{3}+\pi=\frac{5\pi}{3}$ $\displaystyle x-\pi=120^{\circ}\Rightarrow x-\pi=\frac{2\pi}{3}\Rightarrow x=\frac{2\pi}{3}+\pi=\frac{5\pi}{3}$

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Example Question #2 : Radians And Conversions

Give $\displaystyle x$ in degrees:

$\frac{3x+6\pi}{2}=4\pi$ $\displaystyle \frac{3x+6\pi}{2}=4\pi$

Possible Answers:

$x=150^{\circ}$ $\displaystyle x=150^{\circ}$

$x=210^{\circ}$ $\displaystyle x=210^{\circ}$

$x=180^{\circ}$ $\displaystyle x=180^{\circ}$

$x=30^{\circ}$ $\displaystyle x=30^{\circ}$

$x=120^{\circ}$ $\displaystyle x=120^{\circ}$

Correct answer:

$x=120^{\circ}$ $\displaystyle x=120^{\circ}$

Explanation:

First we can find $\displaystyle x$ in radians:

$\frac{3x+6\pi}{2}=4\pi\Rightarrow 3x+6\pi=2\times 4\pi$ $\displaystyle \frac{3x+6\pi}{2}=4\pi\Rightarrow 3x+6\pi=2\times 4\pi$

$\Rightarrow 3x+6\pi=8\pi\Rightarrow 3x=8\pi-6\pi$ $\displaystyle \Rightarrow 3x+6\pi=8\pi\Rightarrow 3x=8\pi-6\pi$

$\Rightarrow 3x=2\pi\Rightarrow x=\frac{2\pi}{3}$ $\displaystyle \Rightarrow 3x=2\pi\Rightarrow x=\frac{2\pi}{3}$

To change radians to degrees we need to multiply radians by $\frac{180^{\circ}}{\pi}$ $\displaystyle \frac{180^{\circ}}{\pi}$. So we can write:

$x=\frac{2\pi}{3}\Rightarrow x=\frac{2\pi}{3}\times \frac{180^{\circ}}{\pi}=120^{\circ}$ $\displaystyle x=\frac{2\pi}{3}\Rightarrow x=\frac{2\pi}{3}\times \frac{180^{\circ}}{\pi}=120^{\circ}$

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Example Question #4 : Radians

Give $\displaystyle x$ in degrees:

$\frac{4x-\pi}{2}=3\pi$ $\displaystyle \frac{4x-\pi}{2}=3\pi$

Possible Answers:

$x=345^{\circ}$ $\displaystyle x=345^{\circ}$

$x=305^{\circ}$ $\displaystyle x=305^{\circ}$

$x=315^{\circ}$ $\displaystyle x=315^{\circ}$

$x=415^{\circ}$ $\displaystyle x=415^{\circ}$

$x=300^{\circ}$ $\displaystyle x=300^{\circ}$

Correct answer:

$x=315^{\circ}$ $\displaystyle x=315^{\circ}$

Explanation:

First we can find $\displaystyle x$ in radians:

$\frac{4x-\pi}{2}=3\pi\Rightarrow 4x-\pi=2\times 3\pi$ $\displaystyle \frac{4x-\pi}{2}=3\pi\Rightarrow 4x-\pi=2\times 3\pi$

$\Rightarrow 4x-\pi=6\pi\Rightarrow 4x=6\pi+\pi$ $\displaystyle \Rightarrow 4x-\pi=6\pi\Rightarrow 4x=6\pi+\pi$

$\Rightarrow 4x=7\pi\Rightarrow x=\frac{7\pi}{4}$ $\displaystyle \Rightarrow 4x=7\pi\Rightarrow x=\frac{7\pi}{4}$

To change radians to degrees we need to multiply radians by $\frac{180^{\circ}}{\pi}$ $\displaystyle \frac{180^{\circ}}{\pi}$. So we can write:

$x=\frac{7\pi}{4}\Rightarrow x=\frac{7\pi}{4}\times \frac{180^{\circ}}{\pi}=315^{\circ}$ $\displaystyle x=\frac{7\pi}{4}\Rightarrow x=\frac{7\pi}{4}\times \frac{180^{\circ}}{\pi}=315^{\circ}$

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Example Question #3 : Radians And Conversions

Give $\displaystyle x$ in radians:

$\frac{72^{\circ}-\pi}{2x-\pi}=3$ $\displaystyle \frac{72^{\circ}-\pi}{2x-\pi}=3$

Possible Answers:

$x=\frac{2\pi}{5}$ $\displaystyle x=\frac{2\pi}{5}$

$x=\frac{\pi}{5}$ $\displaystyle x=\frac{\pi}{5}$

$x=\frac{4\pi}{5}$ $\displaystyle x=\frac{4\pi}{5}$

$x=\frac{2\pi}{3}$ $\displaystyle x=\frac{2\pi}{3}$

$x=\frac{3\pi}{5}$ $\displaystyle x=\frac{3\pi}{5}$

Correct answer:

$x=\frac{2\pi}{5}$ $\displaystyle x=\frac{2\pi}{5}$

Explanation:

First we need to convert $72^{\circ}$ $\displaystyle 72^{\circ}$ to radians by multiplying by $\frac{\pi}{180^{\circ}}$ $\displaystyle \frac{\pi}{180^{\circ}}$:

$72^{\circ}\times \frac{\pi}{180^{\circ}}=\frac{2\pi}{5}$ $\displaystyle 72^{\circ}\times \frac{\pi}{180^{\circ}}=\frac{2\pi}{5}$

Now we can solve the following equation for $\displaystyle x$:

$\frac{72^{\circ}-\pi}{2x-\pi}=3\Rightarrow \frac{\frac{2\pi}{5}-\pi}{2x-\pi}=3$ $\displaystyle \frac{72^{\circ}-\pi}{2x-\pi}=3\Rightarrow \frac{\frac{2\pi}{5}-\pi}{2x-\pi}=3$

$\Rightarrow {\frac{2\pi}{5}-\pi}=3(2x-\pi)\Rightarrow \frac{2\pi-5\pi}{5}=6x-3\pi$ $\displaystyle \Rightarrow {\frac{2\pi}{5}-\pi}=3(2x-\pi)\Rightarrow \frac{2\pi-5\pi}{5}=6x-3\pi$

$\Rightarrow -3\pi=30x-15\pi\Rightarrow 30x=12\pi$ $\displaystyle \Rightarrow -3\pi=30x-15\pi\Rightarrow 30x=12\pi$

$\Rightarrow x=\frac{12\pi}{30}\Rightarrow x=\frac{2\pi}{5}$ $\displaystyle \Rightarrow x=\frac{12\pi}{30}\Rightarrow x=\frac{2\pi}{5}$

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Example Question #3 : Radians And Conversions

Give $\displaystyle x$ in degrees:

$\frac{\frac{\pi}{2}-x}{\frac{\pi}{2}+x}=\frac{1}{2}$ $\displaystyle \frac{\frac{\pi}{2}-x}{\frac{\pi}{2}+x}=\frac{1}{2}$

Possible Answers:

$x=60^{\circ}$ $\displaystyle x=60^{\circ}$

$x=90^{\circ}$ $\displaystyle x=90^{\circ}$

$x=30^{\circ}$ $\displaystyle x=30^{\circ}$

$x=36^{\circ}$ $\displaystyle x=36^{\circ}$

$x=40^{\circ}$ $\displaystyle x=40^{\circ}$

Correct answer:

$x=30^{\circ}$ $\displaystyle x=30^{\circ}$

Explanation:

First we can find $\displaystyle x$ in radians:

$\frac{\frac{\pi}{2}-x}{\frac{\pi}{2}+x}=\frac{1}{2}\Rightarrow 2(\frac{\pi}{2}-x)=1(\frac{\pi}{2}+x)$ $\displaystyle \frac{\frac{\pi}{2}-x}{\frac{\pi}{2}+x}=\frac{1}{2}\Rightarrow 2(\frac{\pi}{2}-x)=1(\frac{\pi}{2}+x)$

$\Rightarrow \pi-2x=\frac{\pi}{2}+x\Rightarrow \pi-\frac{\pi}{2}=3x$ $\displaystyle \Rightarrow \pi-2x=\frac{\pi}{2}+x\Rightarrow \pi-\frac{\pi}{2}=3x$

$\Rightarrow 3x=\frac{\pi}{2}\Rightarrow x=\frac{\pi}{6}$ $\displaystyle \Rightarrow 3x=\frac{\pi}{2}\Rightarrow x=\frac{\pi}{6}$

To change radians to degrees we need to multiply radians by $\frac{180^{\circ}}{\pi}$ $\displaystyle \frac{180^{\circ}}{\pi}$. So we can write:

$x=\frac{\pi}{6}\Rightarrow x=\frac{\pi}{6}\times \frac{180^{\circ}}{\pi}=30^{\circ}$ $\displaystyle x=\frac{\pi}{6}\Rightarrow x=\frac{\pi}{6}\times \frac{180^{\circ}}{\pi}=30^{\circ}$

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Example Question #3 : Radians And Conversions

Convert the angle $\frac{7\pi }{3}$ $\displaystyle \frac{7\pi }{3}$ into degrees.

Possible Answers:

$420^{\circ}$ $\displaystyle 420^{\circ}$

$315^{\circ}$ $\displaystyle 315^{\circ}$

$840^{\circ}$ $\displaystyle 840^{\circ}$

$500^{\circ}$ $\displaystyle 500^{\circ}$

$210^{\circ}$ $\displaystyle 210^{\circ}$

Correct answer:

$420^{\circ}$ $\displaystyle 420^{\circ}$

Explanation:

To convert radians to degrees, use the conversion $\frac{180^{\circ}}{\pi }$ $\displaystyle \frac{180^{\circ}}{\pi }$.

In this case:

$(\frac{7\pi }{3})*(\frac{180^{\circ}}{\pi }) = 420^{\circ}$ $\displaystyle (\frac{7\pi }{3})*(\frac{180^{\circ}}{\pi }) = 420^{\circ}$

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Example Question #21 : Unit Circle And Radians

How many radians are in $300^\circ$ $\displaystyle 300^\circ$?

Possible Answers:

$\frac{5\pi}{3}$ $\displaystyle \frac{5\pi}{3}$

$\frac{7\pi}{3}$ $\displaystyle \frac{7\pi}{3}$

$\frac{3\pi}{5}$ $\displaystyle \frac{3\pi}{5}$

$\pi\sqrt{3}$ $\displaystyle \pi\sqrt{3}$

$\frac{2\pi}{3}$ $\displaystyle \frac{2\pi}{3}$

Correct answer:

$\frac{5\pi}{3}$ $\displaystyle \frac{5\pi}{3}$

Explanation:

Since $180^\circ=\pi\ \text{radians}$ $\displaystyle 180^\circ=\pi\ \text{radians}$, we can solve by setting up a proportion:

$\frac{300^\circ}{x}=\frac{180^\circ}{\pi}$ $\displaystyle \frac{300^\circ}{x}=\frac{180^\circ}{\pi}$

Cross-multiply and solve.

$300^\circ*\pi=180^\circ*x$ $\displaystyle 300^\circ*\pi=180^\circ*x$

$\frac{300^\circ}{180^\circ}\pi=x$ $\displaystyle \frac{300^\circ}{180^\circ}\pi=x$

$\frac{5\pi}{3}=x$ $\displaystyle \frac{5\pi}{3}=x$

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Example Question #4 : Radians And Conversions

How many degrees are in $\frac{5\pi}{6}$ $\displaystyle \frac{5\pi}{6}$ radians?

Possible Answers:

$150^\circ$ $\displaystyle 150^\circ$

$130^\circ$ $\displaystyle 130^\circ$

$210^\circ$ $\displaystyle 210^\circ$

$180^\circ$ $\displaystyle 180^\circ$

$160^\circ$ $\displaystyle 160^\circ$

Correct answer:

$150^\circ$ $\displaystyle 150^\circ$

Explanation:

Since $180^\circ=\pi\ \text{radians}$ $\displaystyle 180^\circ=\pi\ \text{radians}$, we can solve by setting up a proportion:

$\frac{x}{\frac{5\pi}{6}}=\frac{180^\circ}{\pi}$ $\displaystyle \frac{x}{\frac{5\pi}{6}}=\frac{180^\circ}{\pi}$

Cross multiply and solve.

$180^\circ*\frac{5\pi}{6}=x*\pi$ $\displaystyle 180^\circ*\frac{5\pi}{6}=x*\pi$

$\frac{900\pi}{6}=x*\pi$ $\displaystyle \frac{900\pi}{6}=x*\pi$

$150\pi=x*\pi$ $\displaystyle 150\pi=x*\pi$

$\frac{150\pi}{\pi}=x$ $\displaystyle \frac{150\pi}{\pi}=x$

$150^\circ=x$ $\displaystyle 150^\circ=x$

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Trigonometry : Radians and Conversions

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #1 : Radians And Conversions

Example Question #2 : Radians And Conversions

Example Question #1 : Radians And Conversions

Example Question #2 : Radians And Conversions

Example Question #4 : Radians

Example Question #3 : Radians And Conversions

Example Question #3 : Radians And Conversions

Example Question #3 : Radians And Conversions

Example Question #21 : Unit Circle And Radians

Example Question #4 : Radians And Conversions

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