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Radians (Edexcel IGCSE Further Pure Maths): Flashcards

Exam code: 4PM1

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FrontRadian Measure
Define radian.

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Cards in this collection (13)

Define radian.
One radian is the angle at the centre of a circle sector whose arc length is equal to its radius.
Since the whole circumference is $2 π r$ , a full turn is $2 π$ radians.
Complete the two basic radian equivalents:
$2 π radians =______° and π radians =______°$
The completed equivalents are:
$2 π radians = 360 ° and π radians = 180 °$
Everything else follows from these, so $\frac{π}{2}$ is $90 °$ and $\frac{π}{6}$ is $30 °$ .
How do you convert between degrees and radians?
Multiply by $\frac{π}{180}$ to go from degrees to radians, and by $\frac{180}{π}$ to go the other way.
Each is just $π radians = 180 °$ rearranged, so that one fact is all you need to remember.
Convert $\frac{5 π}{4}$ radians to degrees.
Multiply by $\frac{180}{π}$ , giving $\frac{5 π}{4} \times \frac{180}{π} = \frac{5}{4} \times 180 = 225 °$ .
Recognising that $\frac{π}{4}$ is $45 °$ gets there faster, since $5 \times 45 = 225$ .
True or False?
An angle in radians must always be written with a unit symbol.
False.
When $π$ appears in the angle it is obviously in radians, so no symbol is needed; otherwise rad is usually written.
The degree symbol is different: that must never be left off.
When must you work in radians rather than degrees?
Whenever calculus is being done with trigonometric functions.
The derivative and integral results for $\sin$ and $\cos$ only hold when the angle is measured in radians.
For $θ$ in radians, complete the arc length and sector area formulas:
$l =______θ and A = \frac{1}{2}______θ$
The completed formulas are:
$l = r θ and A = \frac{1}{2} r^{2} θ$
The area carries $r^{2}$ because an area is two-dimensional, while an arc is a length and needs only one factor of $r$ .
What are the arc length and sector area formulas when $θ$ is measured in degrees?
They are $l = \frac{θ}{360} \times 2 π r$ and $A = \frac{θ}{360} \times π r^{2}$ .
Each is simply a fraction of the whole circle, that fraction being the angle divided by $360$ .
Why are the radian versions of these formulas so much simpler?
Because $π$ is already built into the radian measure, so no conversion factor is needed at all.
A radian is defined so that the arc length simply is $r θ$ , with nothing left to scale.
Define minor sector and major sector.
A minor sector has an angle at the centre of less than $180 °$ , and a major sector one of more than $180 °$ .
The same words describe the arcs they cut off, so the minor arc is the shorter of the two.
What makes up the perimeter of a sector?
The arc plus the two radii, so for $θ$ in radians it is $r θ + 2 r$ .
Forgetting the two straight edges is the commonest slip here: an arc length on its own is not a perimeter.
Find the exact perimeter of a sector of radius $7$ with angle $\frac{π}{6}$ radians.
The arc measures $7 \times \frac{π}{6} = \frac{7 π}{6}$ , and the two radii add another $14$ .
So the exact perimeter is $\frac{7 π}{6} + 14$ , left in terms of $π$ rather than rounded.
True or False?
Doubling the radius of a sector doubles its arc length but quadruples its area.
True.
The arc length $r θ$ has a single factor of $r$ , so it doubles, while the area $\frac{1}{2} r^{2} θ$ has $r$ squared, so it grows four times.
That is the ordinary difference between a length and an area, and it holds for any fixed angle.

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