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Exercise 3.1 · Q1

Q.Find the radian measures corresponding to the following degree measures:

(i) 25∘25^\circ
(ii) −47∘30′-47^\circ 30'
(iii) 240∘240^\circ
(iv) 520∘520^\circ
Rajasthan RbseTextbookSubjective· 2mImportance★★★★★est
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✓ Free question

To convert degrees to radians, multiply by π180\frac{\pi}{180}. The results are: (i) 5π36\frac{5\pi}{36},

(ii) −19π72-\frac{19\pi}{72},

(iii) 4π3\frac{4\pi}{3},

(iv) 26π9\frac{26\pi}{9}.

The core idea is simple: degrees and radians are just two different rulers for measuring the same thing — angles. The conversion factor comes from the fact that a full circle is 360∘360^\circ in degrees and 2π2\pi radians in radians. So 360∘=2π360^\circ = 2\pi rad, which simplifies to 180∘=π180^\circ = \pi rad. That single equality is the key.

From 180∘=π180^\circ = \pi rad, we get the conversion factor: 1∘=π1801^\circ = \frac{\pi}{180} rad. So to convert any angle from degrees to radians, you multiply the degree measure by π180\frac{\pi}{180}. That's the entire recipe.

Watch out

A common mistake is to use 180π\frac{180}{\pi} instead of π180\frac{\pi}{180}. Remember: degrees are larger than radians (one degree is about 0.0175 rad), so when you convert degrees to radians, the number should get smaller. Multiplying by π180\frac{\pi}{180} (which is about 0.0175) does exactly that.

Now let's apply this to each case.

  1. For 25∘25^\circ Multiply by π180\frac{\pi}{180}:

25∘=25×π180 rad=25π180 rad25^\circ = 25 \times \frac{\pi}{180} \text{ rad} = \frac{25\pi}{180} \text{ rad}

Simplify the fraction. Both 25 and 180 are divisible by 5:

25π180=5π36 rad\frac{25\pi}{180} = \frac{5\pi}{36} \text{ rad}

So 25∘25^\circ is 5π36\frac{5\pi}{36} radians.

  1. For −47∘30′-47^\circ 30' The negative sign just means the angle is measured clockwise; the conversion works the same way. First, handle the minutes. 30′30' means 30 minutes of arc, and 1∘=60′1^\circ = 60', so 30′=3060∘=0.5∘30' = \frac{30}{60}^\circ = 0.5^\circ. Therefore, −47∘30′=−47.5∘-47^\circ 30' = -47.5^\circ. Now convert:

−47.5∘=−47.5×π180 rad=−47.5π180 rad-47.5^\circ = -47.5 \times \frac{\pi}{180} \text{ rad} = -\frac{47.5\pi}{180} \text{ rad}

To avoid decimals, write 47.5=95247.5 = \frac{95}{2}:

−952×π180=−95π360-\frac{95}{2} \times \frac{\pi}{180} = -\frac{95\pi}{360}

Simplify by dividing numerator and denominator by 5:

−95π360=−19π72 rad-\frac{95\pi}{360} = -\frac{19\pi}{72} \text{ rad}

So −47∘30′-47^\circ 30' is −19π72-\frac{19\pi}{72} radians.

  1. For 240∘240^\circ Multiply:

240∘=240×π180 rad=240π180 rad240^\circ = 240 \times \frac{\pi}{180} \text{ rad} = \frac{240\pi}{180} \text{ rad}

Simplify. Both 240 and 180 are divisible by 60:

240π180=4π3 rad\frac{240\pi}{180} = \frac{4\pi}{3} \text{ rad}

So 240∘240^\circ is 4π3\frac{4\pi}{3} radians.

  1. For 520∘520^\circ Multiply:

520∘=520×π180 rad=520π180 rad520^\circ = 520 \times \frac{\pi}{180} \text{ rad} = \frac{520\pi}{180} \text{ rad}

Simplify. Divide numerator and denominator by 20:

520π180=26π9 rad\frac{520\pi}{180} = \frac{26\pi}{9} \text{ rad}

So 520∘520^\circ is 26π9\frac{26\pi}{9} radians.

Tip

When simplifying, always look for the greatest common divisor (GCD) of the degree number and 180. For 520 and 180, the GCD is 20, so dividing both by 20 gives the simplest fraction immediately.

✓Final answer

The radian measures are: (i) 5π36\boxed{\frac{5\pi}{36}},

(ii) −19π72\boxed{-\frac{19\pi}{72}},

(iii) 4π3\boxed{\frac{4\pi}{3}},

(iv) 26π9\boxed{\frac{26\pi}{9}}.

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