Radians and Conversions - Trigonometry

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Question

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

Answer

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

Therefore, to solve:

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

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Question

Change $\frac{\pi}{5}$ angle to degrees.

Answer

In order to change radians to degrees, we need to multiply the radian agle measure by $\frac{180^{\circ}}{\pi}$ .

$\frac{\pi}{5}\times \frac{180^{\circ}}{\pi}=\frac{180^{\circ}}{5}=36^{\circ}$

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Question

An angle of 40 radians is equal to how many degrees?

Answer

We know that one radian is equal to $\frac{180^{\circ}}{\pi}$ , so in order to change radians to degrees we need to multiply by $\frac{180^{\circ}}{\pi}$ .

$40\times \frac{180^{\circ}}{\pi}=\frac{7200^{\circ}}{\pi}$

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Question

Change $\frac{17\pi}{5}$ angle to degrees.

Answer

In order to change radians to degrees we need to multiply the radians by $\frac{180^{\circ}}{\pi}$ .

$\frac{17\pi}{5}\times \frac{180^{\circ}}{\pi}=\frac{17\times 180^{\circ}}{5}=612^{\circ}$

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Question

Change $75^{\circ}$ to radians.

Answer

In order to change degrees to radians we need to multiply the degrees by $\frac{\pi}{180^{\circ}}$ .

$75^{\circ}\times \frac{\pi}{180^{\circ}}=\frac{5\pi}{12}$

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Question

Change the following expression to degrees:

$\frac{\frac{\pi}{3}-\frac{\pi}{6}}{\frac{1}{3}+\frac{1}{6}}$

Answer

First we need to simplify the expression:

$\frac{\frac{\pi}{3}-\frac{\pi}{6}}{\frac{1}{3}+\frac{1}{6}}=\frac{\frac{2\pi-\pi}{6}}{\frac{2+1}{6}}=\frac{\pi}{3}$

Now multiply by $\frac{180^{\circ}}{\pi}$ :

$\frac{\pi}{3}\times \frac{180^{\circ}}{\pi}=\frac{180^{\circ}}{3}=60^{\circ}$

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Question

Change the following expression to degrees:

${\frac{\pi}{4}-\frac{\pi}{12}}$

Answer

First we need to simplify the expression:

${\frac{\pi}{4}-\frac{\pi}{12}}=\frac{3\pi-\pi}{12}=\frac{2\pi}{12}=\frac{\pi}{6}$

Then multiply the radians by $\frac{180^{\circ}}{\pi}$ :

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

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Question

Convert the following expression to radians:

$\frac{45^{\circ}+30^{\circ}}{45-30}$

Answer

First we need to simplify the expression:

$\frac{45^{\circ}+30^{\circ}}{45-30}=\frac{75^{\circ}}{15}=5^{\circ}$

In order to change degrees to radians, we need to multiply by $\frac{\pi}{180^{\circ}}$ :

$5^{\circ}\times \frac{\pi}{180^{\circ}}=\frac{\pi}{36}$

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Question

Change $270^{\circ}$ to radians.

Answer

In order to change degrees to radians we need to multiply degrees by $\frac{\pi}{180^{\circ}}$ :

$270^{\circ}\times \frac{\pi}{180^{\circ}}=\frac{3\pi}{2}=1.5\pi$

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Question

Convert one radian to degrees.

Answer

In order to change radians to degrees we need to multiply radians by $\frac{180^{\circ}}{\pi}$ :

$1\times \frac{180^{\circ}}{\pi}=\frac{180^{\circ}}{\pi}\approx 57.3^{\circ}$

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Question

Convert one degree to radians.

Answer

In order to change degrees to radians we need to multiply the degrees by $\frac{\pi}{180^{\circ}}$ :

$1^{\circ}\times \frac{\pi}{180^{\circ}}=\frac{\pi}{180}$

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Question

Simplify and give the followoing expression in degrees:

$2\pi-\frac{\pi}{4}$

Answer

First we need to simplify the expression:

$2\pi-\frac{\pi}{4}=\frac{8\pi-\pi}{4}=\frac{7\pi}{4}$

Then multiply by $\frac{180^{\circ}}{\pi}$ :

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

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Question

Simplify and give the followoing expression in degrees:

$\frac{\frac{2\pi}{7}+\frac{\pi}{14}}{\frac{1}{7}}$

Answer

First we need to simplify the expression:

$\frac{\frac{2\pi}{7}+\frac{\pi}{14}}{\frac{1}{7}}=\frac{\frac{4\pi+\pi}{14}}{\frac{1}{7}}=\frac{5\pi}{2}$

Then multiply by $\frac{180^{\circ}}{\pi}$ :

$\frac{5\pi}{2}\times \frac{180^{\circ}}{\pi}=\frac{5\times 180^{\circ}}{2}=450^{\circ}$

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Question

Give in radians:

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

Answer

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

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

Now we can write:

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

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

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

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

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Question

Give in radians:

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

Answer

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

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

Now we can write:

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

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Question

Give in degrees:

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

Answer

First we can find in radians:

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

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

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

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

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

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Question

Give in degrees:

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

Answer

First we can find in radians:

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

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

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

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

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

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Question

Give in radians:

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

Answer

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

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

Now we can solve the following equation for :

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

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

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

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

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Question

Give in degrees:

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

Answer

First we can find in radians:

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

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

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

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

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

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Question

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

Answer

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

In this case:

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

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