$ ext{Express }135^{\circ} ext{ in radians :}$

Q: $\text{Express }135^{\circ} \text{ in radians :}$

To express $135^{\circ}$ in radians, we use the conversion factor between degrees and radians. The relationship between degrees and radians is given by: \[ 180^{\circ} = \pi \text{ radians} \] To convert degrees to radians, we use the formula: \[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180^{\circ}} \] Applying this formula to $135^{\circ}$: \[ 135^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{135\pi}{180} \] Simplifying $\frac{135\pi}{180}$: 1. Find the greatest common divisor (GCD) of 135 and 180, which is 45. 2. Divide both the numerator and the denominator by 45: \[ \frac{135\pi}{180} = \frac{135 \div 45 \cdot \pi}{180 \div 45} = \frac{3\pi}{4} \] Thus, $135^{\circ}$ is equivalent to $\frac{3\pi}{4}$ radians, which corresponds to Option 2. **Explanation for why each of the other options is incorrect:** 1. **Option 1: $\frac{2\pi}{3}$** - $\frac{2\pi}{3}$ radians corresponds to $120^{\circ}$, not $135^{\circ}$. This is calculated by converting $\frac{2\pi}{3}$ radians back to degrees: \[ \frac{2\pi}{3} \times \frac{180^{\circ}}{\pi} = 120^{\circ} \] 2. **Option 3: $\frac{2\pi}{5}$** - $\frac{2\pi}{5}$ radians corresponds to $72^{\circ}$, not $135^{\circ}$. This is calculated by converting $\frac{2\pi}{5}$ radians back to degrees: \[ \frac{2\pi}{5} \times \frac{180^{\circ}}{\pi} = 72^{\circ} \] 3. **Option 4: $\frac{3\pi}{5}$** - $\frac{3\pi}{5}$ radians corresponds to $108^{\circ}$, not $135^{\circ}$. This is calculated by converting $\frac{3\pi}{5}$ radians back to degrees: \[ \frac{3\pi}{5} \times \frac{180^{\circ}}{\pi} = 108^{\circ} \] Therefore, the correct conversion of $135^{\circ}$ to radians is $\frac{3\pi}{4}$, which is Option 2.

Question

$	ext{Express }135^{\circ} 	ext{ in radians :}$

Accepted Answer

To express $135^{\circ}$ in radians, we use the conversion factor between degrees and radians. The relationship between degrees and radians is given by:

$$ 180^{\circ} = \pi \text{ radians} $$

To convert degrees to radians, we use the formula:

$$ \text{Radians} = \text{Degrees} \times \frac{\pi}{180^{\circ}} $$

Applying this formula to $135^{\circ}$:

$$ 135^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{135\pi}{180} $$

Simplifying $\frac{135\pi}{180}$:

1. Find the greatest common divisor (GCD) of 135 and 180, which is 45.
2. Divide both the numerator and the denominator by 45:

$$ \frac{135\pi}{180} = \frac{135 \div 45 \cdot \pi}{180 \div 45} = \frac{3\pi}{4} $$

Thus, $135^{\circ}$ is equivalent to $\frac{3\pi}{4}$ radians, which corresponds to Option 2.

**Explanation for why each of the other options is incorrect:**

1. **Option 1: $\frac{2\pi}{3}$**

- $\frac{2\pi}{3}$ radians corresponds to $120^{\circ}$, not $135^{\circ}$. This is calculated by converting $\frac{2\pi}{3}$ radians back to degrees:
     $$ \frac{2\pi}{3} \times \frac{180^{\circ}}{\pi} = 120^{\circ} $$

2. **Option 3: $\frac{2\pi}{5}$**

- $\frac{2\pi}{5}$ radians corresponds to $72^{\circ}$, not $135^{\circ}$. This is calculated by converting $\frac{2\pi}{5}$ radians back to degrees:
     $$ \frac{2\pi}{5} \times \frac{180^{\circ}}{\pi} = 72^{\circ} $$

3. **Option 4: $\frac{3\pi}{5}$**

- $\frac{3\pi}{5}$ radians corresponds to $108^{\circ}$, not $135^{\circ}$. This is calculated by converting $\frac{3\pi}{5}$ radians back to degrees:
     $$ \frac{3\pi}{5} \times \frac{180^{\circ}}{\pi} = 108^{\circ} $$

Therefore, the correct conversion of $135^{\circ}$ to radians is $\frac{3\pi}{4}$, which is Option 2.