Please solve my question urgently. 110°30′ = __ radians. Solve in full detail
The Correct Answer and Explanation is:
To convert 110∘30′110^\circ 30’110∘30′ into radians, we need to follow a two-step process. Here’s a detailed explanation:
Step 1: Break the angle into degrees and minutes
First, let’s separate the given angle into two parts:
- 110∘110^\circ110∘ (degrees)
- 30′30’30′ (minutes)
Since 1 minute =160= \frac{1}{60}=601 of a degree, we can convert the 30 minutes into a fraction of a degree: 30′=3060=0.5∘30′ = \frac{30}{60} = 0.5^\circ30′=6030=0.5∘
Thus, the total angle in degrees is: 110∘+0.5∘=110.5∘110^\circ + 0.5^\circ = 110.5^\circ110∘+0.5∘=110.5∘
Step 2: Convert degrees to radians
To convert degrees to radians, we use the following formula: radians=degrees×π180\text{radians} = \text{degrees} \times \frac{\pi}{180}radians=degrees×180π
Now, plug in 110.5∘110.5^\circ110.5∘ into the formula: radians=110.5×π180\text{radians} = 110.5 \times \frac{\pi}{180}radians=110.5×180π
Step 3: Perform the calculation
Now, calculate the value: 110.5π180=110.5180×π=0.613888×π\frac{110.5 \pi}{180} = \frac{110.5}{180} \times \pi = 0.613888 \times \pi180110.5π=180110.5×π=0.613888×π
Using the approximate value of π≈3.1416\pi \approx 3.1416π≈3.1416: 0.613888×3.1416≈1.928 radians0.613888 \times 3.1416 \approx 1.928 \, \text{radians}0.613888×3.1416≈1.928radians
Thus, the angle 110∘30′110^\circ 30’110∘30′ is approximately equal to: 1.928 radians\boxed{1.928 \, \text{radians}}1.928radians
Explanation of the Process:
- Degrees to Radians Formula: The conversion from degrees to radians is done by multiplying the degree measure by π180\frac{\pi}{180}180π, because π\piπ radians corresponds to 180∘180^\circ180∘.
- Minutes Conversion: Minutes are a subdivision of degrees. Each minute is 160\frac{1}{60}601 of a degree, which allows us to convert the minutes into a fraction of a degree and then add it to the degree measure.
- Final Calculation: The calculation involves straightforward multiplication, and by approximating π\piπ, we arrive at the radian value.
So, the final answer is approximately 1.9281.9281.928 radians.
