Using Radians - SAT Math

Card 0 of 10

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:

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

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

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

Answer

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

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

Cross multiply and solve.

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

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

$150\pi=x\pi$

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

$150^\circ=x$

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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

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

Simplify and give the following 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

Convert $\frac{7\pi}{4}$ radians into degrees.

Answer

Recall the definition of "radians" derived from the unit circle:

$180$^{\circ}$ = \pi rad$

The quantity of radians given in the problem is $\frac{7\pi}{4}$ . All that is required to convert this measure into degrees is to denote the unknown angle measure in degrees by $\Theta$ and set up a proportion equation using the aforementioned definition relating radians to degrees:

$\frac{180^{\circ}}{\Theta} = \frac{\pi rad}{\frac{7\pi}{4} rad}$

Cross-multiply the denominators in these fractions to obtain:

$1260^{\circ}\pi rad=4\Theta\pi rad$

or

$315^{\circ}\pi rad =\Theta\pi rad$ .

Canceling like terms in these equations yields

$\Theta = 315^{\circ}$

Hence, the correct angle measure of $\frac{7\pi}{4}$ in degrees is $315^{\circ}$ .

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Question

$\frac{37\pi}{18}$ radians is equivalent to how many degrees?

Answer

1 radian is equal to $\frac{180}{\pi}$ degrees. Using this conversion factor,

$\frac{37\pi}{18}\times\frac{180}{\pi}=37\times10=370$ .

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Question

Simplify your answer.

Convert $45^{\circ}$ to radians:

Answer

We know that:

$360^{\circ}=2\pi$ Radians

since the giving angle was in degrees then we multiply

$45^{\circ}*\frac{\pi}{180^{\circ}}= \frac{\pi}{4}$

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Question

Give your answer in terms of $\pi$ .

Convert $165^{\circ}$ to radians:

Answer

To convert degrees to radians, we need to multiply the given degree by $\frac{\pi}{180^{\circ}}$ .

$165*^{\circ}\frac{\pi}{180^{\circ}}=\frac{165\pi}{180}$

To simplify, we get:

$\frac{11\pi}{12}$

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