Mathematics
Grade 12
15 min
Convert between radians and degrees
Convert between radians and degrees
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1
Introduction & Learning Objectives
Learning Objectives
Define a radian and explain its relationship to the radius and arc length of a circle.
Recall and apply the fundamental conversion identity: π radians = 180°.
Derive and use the conversion factors to convert angle measures from degrees to radians.
Derive and use the conversion factors to convert angle measures from radians to degrees.
Express radian measures as exact values in terms of π.
Identify the degree and radian measures of key angles on the unit circle (e.g., 30°, 45°, 60°, 90° and their multiples).
Solve applied problems involving angular velocity that require unit conversion.
Why do mathematicians and physicists prefer a slice of 'pi' for their angles instead of the familiar 360 degrees? 🥧 Let's find out why radians are essenti...
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Key Concepts & Vocabulary
TermDefinitionExample
Degree (°)A unit of angular measure, defined such that one full rotation is equal to 360 degrees.A right angle measures 90°, and a straight line forms an angle of 180°.
Radian (rad)The measure of a central angle of a circle that subtends an arc whose length is equal to the circle's radius. It is a dimensionless unit.If a circle has a radius 'r', an angle of 1 radian cuts off an arc of length 'r'. A full circle is 2π radians.
Unit CircleA circle with a radius of 1 centered at the origin of a Cartesian plane. On the unit circle, the arc length is numerically equal to the central angle measured in radians.An angle of π/2 radians on the unit circle corresponds to an arc length of π/2.
Arc Length (s)The distance along the perimeter of a circle sub...
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Core Formulas
Fundamental Relationship
\pi \text{ radians} = 180^{\circ}
This is the foundational identity that connects the two systems of angle measurement. It states that a half-rotation is equivalent in both measures.
Degrees to Radians Conversion
\text{Radians} = \text{Degrees} \times \frac{\pi}{180^{\circ}}
To convert an angle from degrees to radians, multiply the degree measure by the conversion factor π/180°. The degree units will cancel out.
Radians to Degrees Conversion
\text{Degrees} = \text{Radians} \times \frac{180^{\circ}}{\pi}
To convert an angle from radians to degrees, multiply the radian measure by the conversion factor 180°/π. The π units will often cancel out.
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Challenging
A car is traveling at 90 km/h. Its tires have a radius of 30 cm. What is the angular velocity of the tires in radians per second?
A.8.33 rad/s
B.3000 rad/s
C.250/3 rad/s
D.83.33 rad/s
Challenging
A pizza with a 14-inch diameter is cut into 8 identical slices. What is the area of one slice, in square inches? (Area of a sector = (1/2)r²θ, with θ in radians).
A.7π/4
B.49π/8
C.49π/4
D.7π/2
Challenging
In calculus, the derivative of sin(x) is cos(x) only when x is in radians. This relies on the fundamental limit lim(h→0) sin(h)/h = 1. What property of radians, stemming from its definition, makes this limit true?
A.For small angles, the arc length (s) is approximately equal to the chord length, and both are equal to rθ, making sin(θ) ≈ θ.
B.Radians are based on π, which is an irrational number essential for calculus.
C.The conversion factor 180/π simplifies derivatives more easily than π/180.
D.Radians can be expressed as fractions, whereas degrees are always integers.
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