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Circular Measure (Radians) (Cambridge (CIE) A Level Maths: Pure 1): Flashcards

Exam code: 9709

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FrontRadian Measure
Define radian.

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Cards in this collection (13)

Define radian.
One radian is the angle in a sector whose radius and arc length are both $1$ .
Radians are an alternative to degrees, and are usually quoted in terms of $π$ .
$π$ radians $=______°$ , so a full turn of $2 π$ radians is $______°$ .
$π$ radians $= 180 °$ , so a full turn of $2 π$ radians is $360 °$ .
Every other conversion follows from $π = 180 °$ , so $\frac{3}{4} π = 135 °$ .
What is the arc length of a sector of radius $r$ and angle $θ$ ?
$s = r θ$ , with $θ$ in radians; in degrees the formula simply does not hold.
A full turn, $θ = 2 π$ , gives $s = 2 π r$ , which is the whole circumference.
What is the area of a sector of radius $r$ and angle $θ$ ?
$A = \frac{1}{2} r^{2} θ$ , again with $θ$ in radians.
Both sector formulae are just the whole-circle value scaled by $\frac{θ}{2 π}$ .
True or False?
One radian is a little under $60 °$ .
True.
Since $π$ radians is $180 °$ , one radian is $\frac{180}{π} \approx 57.3 °$ .
Knowing roughly how big a radian is makes a useful check on an answer.
How can you tell an angle is in radians when no symbol is given?
If it is written in terms of $π$ , it is almost always in radians.
Otherwise the symbol $rad$ is used, or occasionally a superscript $c$ .
How do you find the area of the segment cut off by a chord in a circle?
Work out the sector area, then subtract the triangle formed by the two radii and the chord.
That triangle has area $\frac{1}{2} r^{2} \sin θ$ , since both of its known sides are radii.
Where do the exact values for $30 °$ , $45 °$ and $60 °$ come from?
From SOH CAH TOA applied to two special triangles: half an equilateral triangle gives $30 °$ and $60 °$ , and a right-angled isosceles triangle gives $45 °$ .
Remembering the two triangles is safer than remembering six separate values.
$\sin 30 ° =______$ , $\cos 60 ° =______$ , $\tan 45 ° =______$
$\sin 30 ° = \frac{1}{2}$ , $\cos 60 ° = \frac{1}{2}$ , $\tan 45 ° = 1$
The first two agree because $30 °$ and $60 °$ are the two acute angles of the same triangle, with opposite and adjacent swapping over.
What are $30 °$ , $45 °$ and $60 °$ in radians?
$\frac{π}{6}$ , $\frac{π}{4}$ and $\frac{π}{3}$ .
Exact values are quoted in radians as often as in degrees, so both forms need to be recognised on sight.
How do you recall $\sin$ , $\cos$ and $\tan$ of $0 °$ , $90 °$ and $180 °$ ?
Read them off the graphs of the three functions rather than from a triangle.
No triangle can contain an angle of $0 °$ , $90 °$ or $180 °$ , so the triangle method cannot reach them.
How do you find the exact value of $\sin 150 \circ$ ?
Use the symmetry of the graph: $150 °$ mirrors $30 °$ , and sine is positive there, so $\sin 150 ° = \frac{1}{2}$ .
Any multiple of $30 °$ , $45 °$ or $60 °$ can be reached the same way.
True or False?
$\tan 90 °$ has an exact value.
False.
$\tan 90 °$ is undefined, because $\cos 90 ° = 0$ and the tangent divides by it.
The graph has an asymptote there rather than a value.

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