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.