9709 · 1.4
Circular measure flashcards
Revision flashcards for Cambridge 9709 Circular measure (syllabus 1.4). Flip, recall, then mark a real past-paper question.
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What is the definition of one radian?
One radian is the angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle.
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How do you convert from degrees to radians?
Multiply the angle in degrees by $\frac{\pi}{180}$.
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How do you convert from radians to degrees?
Multiply the angle in radians by $\frac{180}{\pi}$.
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What is the formula for the length of an arc?
$s = r\theta$, where $r$ is the radius and $\theta$ is the angle in radians.
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What is the formula for the area of a sector?
$A = \frac{1}{2}r^2\theta$, where $r$ is the radius and $\theta$ is the angle in radians.
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What is the most common mistake when using the arc length and sector area formulae?
Forgetting to ensure the angle $\theta$ is in radians. If the angle is given in degrees, you must convert it first.
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How do you find the area of a segment?
Area of Sector - Area of Triangle. The formula is $A = \frac{1}{2}r^2(\theta - \sin\theta)$ for $\theta$ in radians.
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What is the formula for the perimeter of a sector?
Perimeter = Arc Length + 2 radii. So, $P = r\theta + 2r = r(\theta + 2)$.
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What is the area of a triangle given two sides and the included angle?
$A = \frac{1}{2}ab\sin C$. For a triangle within a sector, this is $A = \frac{1}{2}r^2\sin\theta$.
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Why must your calculator be in 'RAD' mode for these problems?
Because the formulae for arc length, sector area, and segment area are all derived using radians. Also, when calculating $\sin\theta$ for the segment formula, $\theta$ must be in radians.
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What is the exact value of $2\pi$ radians in degrees?
$360^\circ$. This represents a full circle.