Trigonometric Functions, Radian Measure
∏ <-- will be using this for pi
Using exact values, convert to radians ( hint use fraction of ∏)
90° = 90°*∏/180° = 1∏/2
^^would it be right to also call it 2∏ radians?
Determine the angle at the centre of the circle, radius 6cm, for each arc length.Express each angle in radians and then in degrees.
3cm
a = rθ
3 = 6θ
1/2 = θ
1/2*180°/∏
=28.6°
^One of my friends got a totally different answer, and she got
θ = 2 rad
2*180/∏ = 114.59
^Ive no idea how she got that :S
It is obvious she is wrong.
the whole circumference of a 6cm circle is 12pi or about 36 cm
3cm is about 1/12 of the whole, or 1/12 of 360° = 30°
she obviously confused the formula.
ALWAYS do a general sanity check on your answers. Just because you get a value doesn't mean that it's correct.
Thanks so much :)
To convert an angle from degrees to radians, you use the fact that there are π radians in 180 degrees. So, you divide the given angle in degrees by 180 and multiply the result by π to obtain the angle in radians.
For example, to convert 90 degrees to radians:
90° * π/180° = 1π/2 radians
So, your initial conversion of 90 degrees to radians is correct:
90° = 1π/2 radians
However, it is not correct to call it 2π radians. 2π radians is equivalent to 360 degrees, not 90 degrees. Remember that there are only 2π radians in a complete circle.
Now, let's determine the angle at the center of the circle for an arc length of 3 cm with a radius of 6 cm.
We can use the formula a = rθ, where a is the arc length, r is the radius, and θ is the angle at the center in radians.
Given:
a = 3 cm
r = 6 cm
Using the formula, we have:
3 cm = 6 cm * θ
Dividing both sides by 6 cm gives:
1/2 = θ
So, the angle at the center of the circle is 1/2 radian.
To convert it to degrees, you need to use the conversion factor 180/π (180 degrees per π radians):
1/2 radian * 180°/π = 90°/π degrees
This can be left in terms of π, or you can approximate the decimal value using a calculator:
90°/π ≈ 28.6479°
So, your calculation of 28.6° is correct.
As for your friend's answer of θ = 2 radians, it seems she made an error in the calculation. The correct conversion should be:
2 radians * 180°/π ≈ 114.5916°
It appears she used the conversion factor 180/π but made a mistake in the multiplication.
Always double-check the calculations and be careful with the units to ensure accurate results.